### The Mizar article:

### Basic Functions and Operations on Functions

**by****Czeslaw Bylinski**

- Received May 9, 1989
Copyright (c) 1989 Association of Mizar Users

- MML identifier: FUNCT_3
- [ MML identifier index ]

environ vocabulary FUNCT_1, RELAT_1, BOOLE, TARSKI, PARTFUN1, FUNCT_2, FUNCT_3; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, RELSET_1, FUNCT_1, FUNCT_2; constructors TARSKI, FUNCT_2, XBOOLE_0; clusters FUNCT_1, RELSET_1, SUBSET_1, XBOOLE_0, ZFMISC_1; requirements SUBSET, BOOLE; definitions TARSKI; theorems TARSKI, ZFMISC_1, FUNCT_1, FUNCT_2, RELAT_1, RELSET_1, XBOOLE_0, XBOOLE_1; schemes FUNCT_1; begin reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set; reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set; reserve C,C1,C2,D,D1,D2 for non empty set; :: Additional propositions about functions theorem Th1: A c= Y implies id A = (id Y)|A proof assume A c= Y; hence id A = id(Y /\ A) by XBOOLE_1:28 .= (id Y)*(id A) by FUNCT_1:43 .= (id Y)|A by RELAT_1:94; end; theorem Th2: for f,g being Function st X c= dom(g*f) holds f.:X c= dom g proof let f,g be Function such that A1: X c= dom(g*f); let y; assume y in f.:X; then consider x such that x in dom f and A2: x in X and A3: y = f.x by FUNCT_1:def 12; thus thesis by A1,A2,A3,FUNCT_1:21; end; theorem Th3: for f,g being Function st X c= dom f & f.:X c= dom g holds X c= dom(g*f) proof let f,g be Function such that A1: X c= dom f and A2: f.:X c= dom g; let x; assume A3: x in X; then f.x in f.:X by A1,FUNCT_1:def 12; hence thesis by A1,A2,A3,FUNCT_1:21; end; theorem Th4: for f,g being Function st Y c= rng(g*f) & g is one-to-one holds g"Y c= rng f proof let f,g be Function such that A1: Y c= rng(g*f) and A2: g is one-to-one; let y; assume A3: y in g"Y; then g.y in Y by FUNCT_1:def 13; then consider x such that A4: x in dom(g*f) and A5: g.y = (g*f).x by A1,FUNCT_1:def 5; g.y = g.(f.x) & y in dom g & f.x in dom g by A3,A4,A5,FUNCT_1:21,22,def 13; then y = f.x & x in dom f by A2,A4,FUNCT_1:21,def 8; hence y in rng f by FUNCT_1:def 5; end; theorem Th5: for f,g being Function st Y c= rng g & g"Y c= rng f holds Y c= rng(g*f) proof let f,g be Function such that A1: Y c= rng g and A2: g"Y c= rng f; let y; assume A3: y in Y; then consider z such that A4: z in dom g and A5: y = g.z by A1,FUNCT_1:def 5; z in g"Y by A3,A4,A5,FUNCT_1:def 13; then consider x such that A6: x in dom f and A7: z = f.x by A2,FUNCT_1:def 5; x in dom(g*f) & (g*f).x = y by A4,A5,A6,A7,FUNCT_1:21,23; hence thesis by FUNCT_1:def 5; end; scheme FuncEx_3{A()->set,B()-> set,P[set,set,set]}: ex f being Function st dom f = [:A(),B():] & for x,y st x in A() & y in B() holds P[x,y,f.[x,y]] provided A1: for x,y,z1,z2 st x in A() & y in B() & P[x,y,z1] & P[x,y,z2] holds z1 = z2 and A2: for x,y st x in A() & y in B() ex z st P[x,y,z] proof set D = [:A(), B():]; defpred R[set,set] means for x,y st $1=[x,y] holds P[x,y,$2]; A3: for p,y1,y2 st p in D & R[p,y1] & R[p,y2] holds y1 = y2 proof let p,y1,y2; assume p in D; then consider x1,x2 such that A4: x1 in A() & x2 in B() and A5: p = [x1,x2] by ZFMISC_1:def 2; assume (for x,y st p=[x,y] holds P[x,y,y1]) & (for x,y st p=[x,y] holds P[x,y,y2]); then P[x1,x2,y1] & P[x1,x2,y2] by A5; hence thesis by A1,A4; end; A6: for p st p in D ex z st R[p,z] proof let p; assume p in D; then consider x1,y1 such that A7: x1 in A() & y1 in B() and A8: p = [x1,y1] by ZFMISC_1:def 2; consider z such that A9: P[x1,y1,z] by A2,A7; take z; let x,y; assume p=[x,y]; then x=x1 & y=y1 by A8,ZFMISC_1:33; hence thesis by A9; end; consider f being Function such that A10: dom f = D and A11: for p st p in D holds R[p,f.p] from FuncEx(A3,A6); take f; thus dom f = [:A(),B():] by A10; let x,y; assume x in A() & y in B(); then [x,y] in [:A(),B():] by ZFMISC_1:def 2; hence thesis by A11; end; scheme Lambda_3{A()->set,B()->set,F(set,set)->set}: ex f being Function st dom f = [:A(),B():] & for x,y st x in A() & y in B() holds f.[x,y] = F(x,y) proof defpred P[set,set,set] means $3 = F($1,$2); A1: for x,y,z1,z2 st x in A() & y in B() & P[x,y,z1] & P[x,y,z2] holds z1=z2; A2: for x,y st x in A() & y in B() ex z st P[x,y,z]; ex f being Function st dom f = [:A(),B():] & for x,y st x in A() & y in B() holds P[x,y,f.[x,y]] from FuncEx_3(A1,A2); hence thesis; end; theorem Th6: for f,g being Function st dom f = [:X,Y:] & dom g = [:X,Y:] & for x,y st x in X & y in Y holds f.[x,y] = g.[x,y] holds f = g proof let f,g be Function such that A1: dom f = [:X,Y:] & dom g = [:X,Y:] and A2: for x,y st x in X & y in Y holds f.[x,y] = g.[x,y]; p in [:X,Y:] implies f.p = g.p proof assume p in [:X,Y:]; then ex x,y st x in X & y in Y & p = [x,y] by ZFMISC_1:def 2; hence thesis by A2; end; hence thesis by A1,FUNCT_1:9; end; :: Function indicated by the image under a function definition let f be Function; func .:f -> Function means :Def1: dom it = bool dom f & for X st X c= dom f holds it.X = f.:X; existence proof defpred P[set,set] means for X st $1=X holds $2 = f.:X; A1: for x,y1,y2 st x in bool dom f & P[x,y1] & P[x,y2] holds y1=y2 proof let x,y1,y2 such that x in bool dom f and A2: for X st x=X holds y1 = f.:X and A3: for X st x=X holds y2 = f.:X; thus y1 = f.:x by A2 .= y2 by A3; end; A4: for x st x in bool dom f ex y st P[x,y] proof let x such that x in bool dom f; take f.:x; thus thesis; end; consider g being Function such that A5: dom g = bool dom f and A6: for x st x in bool dom f holds P[x,g.x] from FuncEx(A1,A4); take g; thus thesis by A5,A6; end; uniqueness proof let f1,f2 be Function such that A7: dom f1 = bool dom f and A8: for X st X c= dom f holds f1.X = f.:X and A9: dom f2 = bool dom f and A10: for X st X c= dom f holds f2.X = f.:X; for x st x in bool dom f holds f1.x = f2.x proof let x; assume x in bool dom f; then f1.x = f.:x & f2.x = f.:x by A8,A10; hence thesis; end; hence thesis by A7,A9,FUNCT_1:9; end; end; canceled; theorem Th8: for f being Function st X in dom(.:f) holds (.:f).X = f.:X proof let f be Function; assume X in dom(.:f); then X in bool dom f by Def1; hence thesis by Def1; end; theorem for f being Function holds (.:f).{} = {} proof let f be Function; {} c= dom f by XBOOLE_1:2; then (.:f).{} = f.:{} by Def1; hence thesis by RELAT_1:149; end; theorem Th10: for f being Function holds rng(.:f) c= bool rng f proof let f be Function; let y; assume y in rng(.:f); then consider x such that A1: x in dom(.:f) and A2: y = (.:f).x by FUNCT_1:def 5; y = f.:x by A1,A2,Th8; then y c= rng f by RELAT_1:144; hence y in bool rng f; end; canceled; theorem for f being Function holds (.:f).:A c= bool rng f proof let f be Function; (.:f).:A c= rng(.:f) & rng(.:f) c= bool rng f by Th10,RELAT_1:144; hence thesis by XBOOLE_1:1; end; theorem Th13: for f being Function holds (.:f)"B c= bool dom f proof let f be Function; dom(.:f) = bool dom f by Def1; hence thesis by RELAT_1:167; end; theorem for f being Function of X,D holds (.:f)"B c= bool X proof let f be Function of X,D; dom f = X by FUNCT_2:def 1; hence thesis by Th13; end; theorem Th15: for f being Function holds union((.:f).:A) c= f.:(union A) proof let f be Function; let y; assume y in union((.:f).:A); then consider Z such that A1: y in Z and A2: Z in (.:f).:A by TARSKI:def 4; consider X such that A3: X in dom(.:f) and A4: X in A and A5: Z = (.:f).X by A2,FUNCT_1:def 12; y in f.:X by A1,A3,A5,Th8; then consider x such that A6: x in dom f and A7: x in X and A8: y = f.x by FUNCT_1:def 12; x in union A by A4,A7,TARSKI:def 4; hence thesis by A6,A8,FUNCT_1:def 12; end; theorem Th16: for f being Function st A c= bool dom f holds f.:(union A) = union((.:f).:A) proof let f be Function such that A1: A c= bool dom f; A2: union((.:f).:A) c= f.:(union A) by Th15; f.:(union A) c= union((.:f).:A) proof let y; assume y in f.:(union A); then consider x such that x in dom f and A3: x in union A and A4: y = f.x by FUNCT_1:def 12; consider X such that A5: x in X and A6: X in A by A3,TARSKI:def 4; X in bool dom f by A1,A6; then X in dom(.:f) & y in f.:X & (.:f).X = f.:X by A4,A5,Def1,FUNCT_1:def 12; then y in (.:f).X & (.:f).X in (.:f).:A by A6,FUNCT_1:def 12; hence y in union((.:f).:A) by TARSKI:def 4; end; hence thesis by A2,XBOOLE_0:def 10; end; theorem for f being Function of X,D st A c= bool X holds f.:(union A) = union((.:f).:A) proof let f be Function of X,D; dom f = X by FUNCT_2:def 1; hence thesis by Th16; end; theorem Th18: for f being Function holds union((.:f)"B) c= f"(union B) proof let f be Function; let x; assume x in union((.:f)"B); then consider X such that A1: x in X and A2: X in (.:f)"B by TARSKI:def 4; dom(.:f) = bool dom f by Def1; then X in bool dom f & (.:f).X in B by A2,FUNCT_1:def 13; then x in dom f & f.x in f.:X & f.:X in B by A1,Def1,FUNCT_1:def 12; then x in dom f & f.x in union B by TARSKI:def 4; hence x in f"(union B) by FUNCT_1:def 13; end; theorem for f being Function st B c= bool rng f holds f"(union B) = union((.:f)"B) proof let f be Function such that A1: B c= bool rng f; A2: union((.:f)"B) c= f"(union B) by Th18; f"(union B) c= union((.:f)"B) proof let x; assume x in f"(union B); then A3: x in dom f & f.x in union B by FUNCT_1:def 13; then consider Y such that A4: f.x in Y and A5: Y in B by TARSKI:def 4; Y c= rng f & f"Y c= dom f by A1,A5,RELAT_1:167; then f.:(f"Y) = Y & f"Y in bool dom f by FUNCT_1:147; then f"Y in dom(.:f) & (.:f).(f"Y) in B by A5,Def1; then x in f"Y & f"Y in (.:f)"B by A3,A4,FUNCT_1:def 13; hence x in union((.:f)"B) by TARSKI:def 4; end; hence thesis by A2,XBOOLE_0:def 10; end; theorem for f,g being Function holds .:(g*f) = .:g*.:f proof let f,g be Function; for x holds x in dom .:(g*f) iff x in dom(.:g*.:f) proof let x; thus x in dom .:(g*f) implies x in dom(.:g*.:f) proof assume x in dom .:(g*f); then x in bool dom(g*f) by Def1; then x c= dom(g*f) & dom(g*f) c= dom f by RELAT_1:44; then x c= dom f & f.:x c= dom g by Th2,XBOOLE_1:1; then x in bool dom f & f.:x in bool dom g; then x in dom .:f & f.:x in dom .:g by Def1; then x in dom .:f & .:f.x in dom .:g by Th8; hence thesis by FUNCT_1:21; end; assume x in dom(.:g*.:f); then x in dom .:f & .:f.x in dom .:g by FUNCT_1:21; then x in dom .:f & f.:x in dom .:g by Th8; then x in bool dom f & f.:x in bool dom g by Def1; then x c= dom(g*f) by Th3; then x in bool dom(g*f); hence thesis by Def1; end; then A1: dom .:(g*f) = dom(.:g*.:f) by TARSKI:2; for x st x in dom .:(g*f) holds (.:(g*f)).x = (.:g*.:f).x proof let x; assume A2: x in dom .:(g*f); then x in bool dom(g*f) by Def1; then x c= dom(g*f) & dom(g*f) c= dom f by RELAT_1:44; then A3: x c= dom f & f.:x c= dom g by Th2,XBOOLE_1:1; then x in bool dom f & f.:x in bool dom g; then A4: x in dom .:f & f.:x in dom .:g by Def1; thus (.:(g*f)).x = (g*f).:x by A2,Th8 .= g.:(f.:x) by RELAT_1:159 .= .:g.(f.:x) by A3,Def1 .= .:g.(.:f.x) by A3,Def1 .= (.:g*.:f).x by A4,FUNCT_1:23; end; hence thesis by A1,FUNCT_1:9; end; theorem Th21: for f being Function holds .:f is Function of bool dom f, bool rng f proof let f be Function; dom .:f = bool dom f & rng.:f c= bool rng f by Def1,Th10; hence thesis by FUNCT_2:def 1,RELSET_1:11; end; theorem Th22: for f being Function of X,Y st Y = {} implies X = {} holds .:f is Function of bool X, bool Y proof let f be Function of X,Y; assume A1: Y = {} implies X = {}; A2: .:f is Function of bool dom f, bool rng f by Th21; A3: dom f = X & rng f c= Y by A1,FUNCT_2:def 1,RELSET_1:12; A4: dom.:f = bool dom f & rng.:f c= bool rng f by A2,FUNCT_2:def 1,RELSET_1:12; rng f c= Y by RELSET_1:12; then bool rng f c= bool Y by ZFMISC_1:79; then rng.:f c= bool Y by A4,XBOOLE_1:1; hence thesis by A3,A4,FUNCT_2:def 1,RELSET_1:11; end; definition let X,D; let f be Function of X,D; redefine func .:f -> Function of bool X, bool D; coherence by Th22; end; :: Function indicated by the inverse image under a function definition let f be Function; func "f -> Function means :Def2: dom it = bool rng f & for Y st Y c= rng f holds it.Y = f"Y; existence proof defpred P[set,set] means for Y st $1=Y holds $2 = f"Y; A1: for y,x1,x2 st y in bool rng f & P[y,x1] & P[y,x2] holds x1=x2 proof let y,x1,x2 such that y in bool rng f and A2: for Y st y=Y holds x1 = f"Y and A3: for Y st y=Y holds x2 = f"Y; thus x1 = f"y by A2 .= x2 by A3; end; A4: for y st y in bool rng f ex x st P[y,x] proof let y such that y in bool rng f; take f"y; thus thesis; end; consider g being Function such that A5: dom g = bool rng f and A6: for y st y in bool rng f holds P[y,g.y] from FuncEx(A1,A4); take g; thus thesis by A5,A6; end; uniqueness proof let f1,f2 be Function such that A7: dom f1 = bool rng f and A8: for Y st Y c= rng f holds f1.Y = f"Y and A9: dom f2 = bool rng f and A10: for Y st Y c= rng f holds f2.Y = f"Y; for y st y in bool rng f holds f1.y = f2.y proof let y; assume y in bool rng f; then f1.y = f"y & f2.y = f"y by A8,A10; hence thesis; end; hence thesis by A7,A9,FUNCT_1:9; end; end; canceled; theorem Th24: for f being Function st Y in dom("f) holds ("f).Y = f"Y proof let f be Function; dom("f) = bool rng f by Def2; hence thesis by Def2; end; theorem Th25: for f being Function holds rng("f) c= bool dom f proof let f be Function; let x; assume x in rng("f); then consider y such that A1: y in dom("f) and A2: x = "f.y by FUNCT_1:def 5; x = f"y by A1,A2,Th24; then x c= dom f by RELAT_1:167; hence x in bool dom f; end; canceled; theorem for f being Function holds ("f).:B c= bool dom f proof let f be Function; rng("f) c= bool dom f & ("f).:B c= rng("f) by Th25,RELAT_1:144; hence thesis by XBOOLE_1:1; end; theorem for f being Function holds ("f)"A c= bool rng f proof let f be Function; dom("f) = bool rng f by Def2; hence thesis by RELAT_1:167; end; theorem Th29: for f being Function holds union(("f).:B) c= f"(union B) proof let f be Function; let x; assume x in union(("f).:B); then consider X such that A1: x in X and A2: X in ("f).:B by TARSKI:def 4; consider Y such that A3: Y in dom("f) and A4: Y in B and A5: X = "f.Y by A2,FUNCT_1:def 12; Y in bool rng f by A3,Def2; then f.:(f"Y) = Y & ("f).Y = f"Y & f"Y c= dom f by A3,Th24,FUNCT_1:147,RELAT_1:167; then x in dom f & f.x in f.:X & f.:X in B by A1,A4,A5,FUNCT_1:def 12; then x in dom f & f.x in union B by TARSKI:def 4; hence x in f"(union B) by FUNCT_1:def 13; end; theorem for f being Function st B c= bool rng f holds union(("f).:B) = f"(union B) proof let f be Function such that A1: B c= bool rng f; A2: union(("f).:B) c= f"(union B) by Th29; f"(union B) c= union(("f).:B) proof let x; assume x in f"(union B); then A3: x in dom f & f.x in union B by FUNCT_1:def 13; then consider Y such that A4: f.x in Y and A5: Y in B by TARSKI:def 4; Y in bool rng f by A1,A5; then Y in dom("f) & ("f).Y = f"Y by Def2; then x in f"Y & f"Y in ("f).:B by A3,A4,A5,FUNCT_1:def 12,def 13; hence x in union(("f).:B) by TARSKI:def 4; end; hence thesis by A2,XBOOLE_0:def 10; end; theorem Th31: for f being Function holds union(("f)"A) c= f.:(union A) proof let f be Function; let y; assume y in union(("f)"A); then consider Y such that A1: y in Y and A2: Y in ("f)"A by TARSKI:def 4; dom("f) = bool rng f by Def2; then A3: Y in bool rng f & "f.Y in A by A2,FUNCT_1:def 13; then consider x such that A4: x in dom f and A5: y = f.x by A1,FUNCT_1:def 5; A6: Y c= rng f & f"Y in A by A3,Def2; x in f"Y by A1,A4,A5,FUNCT_1:def 13; then x in union A by A6,TARSKI:def 4; hence y in f.:(union A) by A4,A5,FUNCT_1:def 12; end; theorem for f being Function st A c= bool dom f & f is one-to-one holds union(("f)"A) = f.:(union A) proof let f be Function such that A1: A c= bool dom f and A2: f is one-to-one; A3: union(("f)"A) c= f.:(union A) by Th31; f.:(union A) c= union(("f)"A) proof let y; assume y in f.:(union A); then consider x such that A4: x in dom f and A5: x in union A and A6: y = f.x by FUNCT_1:def 12; consider X such that A7: x in X and A8: X in A by A5,TARSKI:def 4; X c= f"(f.:X) & f"(f.:X) c= X & f.:X c= rng f by A1,A2,A8,FUNCT_1:146,152,RELAT_1:144; then f"(f.:X) = X & f.:X in bool rng f by XBOOLE_0:def 10; then ("f).(f.:X) in A & (f.:X) in dom("f) by A8,Def2; then y in (f.:X) & (f.:X) in ("f)"A by A4,A6,A7,FUNCT_1:def 12,def 13; hence y in union(("f)"A) by TARSKI:def 4; end; hence thesis by A3,XBOOLE_0:def 10; end; theorem Th33: for f being Function holds ("f).:B c= (.:f)"B proof let f be Function; let x; assume x in ("f).:B; then consider Y such that A1: Y in dom ("f) and A2: Y in B and A3: x = "f.Y by FUNCT_1:def 12; Y in bool rng f by A1,Def2; then "f.Y = f"Y & Y c= rng f by A1,Th24; then x c= dom f & f.:x in B by A2,A3,FUNCT_1:147,RELAT_1:167; then x in bool dom f & f.:x in B; then x in dom(.:f) & (.:f).x in B by Def1; hence x in (.:f)"B by FUNCT_1:def 13; end; theorem for f being Function st f is one-to-one holds ("f).:B = (.:f)"B proof let f be Function such that A1: f is one-to-one; A2: ("f).:B c= (.:f)"B by Th33; (.:f)"B c= ("f).:B proof let x; assume x in (.:f)"B; then A3: x in dom(.:f) & (.:f).x in B by FUNCT_1:def 13; then x in bool dom f by Def1; then x c= f"(f.:x) & f"(f.:x) c= x & f.:x c= rng f by A1,FUNCT_1:146,152,RELAT_1:144; then x = f"(f.:x) & f.:x in bool rng f by XBOOLE_0:def 10; then x =("f).(f.:x) & f.:x in dom("f) & f.:x in B by A3,Def2,Th8; hence x in ("f).:B by FUNCT_1:def 12; end; hence thesis by A2,XBOOLE_0:def 10; end; theorem Th35: for f being Function,A be set st A c= bool dom f holds ("f)"A c= (.:f).:A proof let f be Function,A be set such that A1: A c= bool dom f; let y; assume y in ("f)"A; then A2: y in dom("f) & "f.y in A by FUNCT_1:def 13; then y in bool rng f by Def2; then y c= rng f & f"y in A by A2,Def2; then f.:(f"y) = y & f"y in A & f"y in bool dom f by A1,FUNCT_1:147; then .:f.(f"y) = y & f"y in A & f"y in dom .:f by Def1; hence y in (.:f).:A by FUNCT_1:def 12; end; theorem Th36: for f being Function,A be set st f is one-to-one holds (.:f).:A c= ("f)"A proof let f be Function,A be set such that A1: f is one-to-one; let y; assume y in (.:f).:A; then consider x such that A2: x in dom(.:f) and A3: x in A and A4: y = .:f.x by FUNCT_1:def 12; x in bool dom f by A2,Def1; then y = f.:x & x c= dom f by A4,Def1; then y c= rng f & f"y c= x & x c= f"y by A1,FUNCT_1:146,152,RELAT_1:144; then y in bool rng f & f"y in A by A3,XBOOLE_0:def 10; then y in dom("f) & "f.y in A by Def2; hence y in ("f)"A by FUNCT_1:def 13; end; theorem for f being Function,A be set st f is one-to-one & A c= bool dom f holds ("f)"A = (.:f).:A proof let f be Function,A be set; assume f is one-to-one & A c= bool dom f; then ("f)"A c= (.:f).:A & (.:f).:A c= ("f)"A by Th35,Th36; hence thesis by XBOOLE_0:def 10; end; theorem for f,g being Function st g is one-to-one holds "(g*f) = "f*"g proof let f,g be Function such that A1: g is one-to-one; for y holds y in dom "(g*f) iff y in dom("f*"g) proof let y; thus y in dom "(g*f) implies y in dom("f*"g) proof assume y in dom "(g*f); then y in bool rng(g*f) by Def2; then y c= rng(g*f) & rng(g*f) c= rng g by RELAT_1:45; then y c= rng g & g"y c= rng f by A1,Th4,XBOOLE_1:1; then y in bool rng g & g"y in bool rng f; then y in dom "g & g"y in dom "f by Def2; then y in dom "g & "g.y in dom "f by Th24; hence thesis by FUNCT_1:21; end; assume y in dom("f*"g); then y in dom "g & "g.y in dom "f by FUNCT_1:21; then y in dom "g & g"y in dom "f by Th24; then y in bool rng g & g"y in bool rng f by Def2; then y c= rng(g*f) by Th5; then y in bool rng(g*f); hence thesis by Def2; end; then A2: dom "(g*f) = dom("f*"g) by TARSKI:2; for y st y in dom "(g*f) holds "(g*f).y = ("f*"g).y proof let y; assume A3: y in dom "(g*f); then y in bool rng(g*f) by Def2; then y c= rng(g*f) & rng(g*f) c= rng g by RELAT_1:45; then A4: y c= rng g & g"y c= rng f by A1,Th4,XBOOLE_1:1; then y in bool rng g & g"y in bool rng f; then A5: y in dom "g & g"y in dom "f by Def2; thus "(g*f).y = (g*f)"y by A3,Th24 .= f"(g"y) by RELAT_1:181 .= "f.(g"y) by A4,Def2 .= "f.("g.y) by A4,Def2 .= ("f*"g).y by A5,FUNCT_1:23; end; hence thesis by A2,FUNCT_1:9; end; theorem for f being Function holds "f is Function of bool rng f, bool dom f proof let f be Function; dom"f = bool rng f & rng"f c= bool dom f by Def2,Th25; hence thesis by FUNCT_2:def 1,RELSET_1:11; end; :: Characteristic function definition let A,X; func chi(A,X) -> Function means :Def3: dom it = X & for x st x in X holds (x in A implies it.x = 1) & (not x in A implies it.x = 0); existence proof defpred P[set,set] means ($1 in A implies $2 = 1) & (not $1 in A implies $2 = 0); A1: for x,y1,y2 st x in X & P[x,y1] & P[x,y2] holds y1 = y2; A2: for x st x in X ex y st P[x,y] proof let x; assume x in X; not x in A implies ex y st y = 0 & (x in A implies y = 1) & (not x in A implies y = 0); hence thesis; end; ex f being Function st dom f = X & for x st x in X holds P[x,f.x] from FuncEx(A1,A2); hence thesis; end; uniqueness proof let f1,f2 be Function such that A3: dom f1 = X and A4: for x st x in X holds (x in A implies f1.x = 1) & (not x in A implies f1.x = 0) and A5: dom f2 = X and A6: for x st x in X holds (x in A implies f2.x = 1) & (not x in A implies f2.x = 0); for x st x in X holds f1.x=f2.x proof let x; assume x in X; then (x in A implies f1.x = 1 & f2.x = 1) & (not x in A implies f1.x = 0 & f2.x = 0) by A4,A6; hence thesis; end; hence thesis by A3,A5,FUNCT_1:9; end; end; canceled 2; theorem Th42: chi(A,X).x = 1 implies x in A proof assume A1: chi(A,X).x = 1; per cases; suppose x in X; hence thesis by A1,Def3; suppose not x in X; then not x in dom chi(A,X) by Def3; hence thesis by A1,FUNCT_1:def 4; end; theorem x in X \ A implies chi(A,X).x = 0 proof assume x in X\A; then x in X & not x in A by XBOOLE_0:def 4; hence thesis by Def3; end; canceled 3; theorem A c= X & B c= X & chi(A,X) = chi(B,X) implies A = B proof assume that A1: A c= X and A2: B c= X and A3: chi(A,X) = chi(B,X); x in A iff x in B proof thus x in A implies x in B proof assume x in A; then chi(A,X).x = 1 by A1,Def3; hence thesis by A3,Th42; end; assume x in B; then chi(B,X).x = 1 by A2,Def3; hence thesis by A3,Th42; end; hence thesis by TARSKI:2; end; theorem Th48: rng chi(A,X) c= {0,1} proof let y; assume y in rng chi(A,X); then consider x such that A1: x in dom chi(A,X) and A2: y = chi(A,X).x by FUNCT_1:def 5; x in X & (x in A or not x in A) by A1,Def3; then y = 0 or y = 1 by A2,Def3; hence thesis by TARSKI:def 2; end; theorem for f being Function of X,{0,1} holds f = chi(f"{1},X) proof let f be Function of X,{0,1}; now thus A1: dom f = X by FUNCT_2:def 1; let x such that A2: x in X; thus x in f"{1} implies f.x = 1 proof assume x in f"{1}; then f.x in {1} by FUNCT_1:def 13; hence thesis by TARSKI:def 1; end; assume not x in f"{1}; then not f.x in {1} & f.x in rng f & rng f c= {0,1} by A1,A2,FUNCT_1:def 5,def 13,RELSET_1:12; then f.x <> 1 & f.x in {0,1} by TARSKI:def 1; hence f.x = 0 by TARSKI:def 2; end; hence thesis by Def3; end; definition let A,X; redefine func chi(A,X) -> Function of X,{0,1}; coherence proof {0,1} <> {} & dom chi(A,X) = X & rng chi(A,X) c= {0,1} by Def3,Th48; hence thesis by FUNCT_2:def 1,RELSET_1:11; end; end; definition let Y; let A be Subset of Y; func incl(A) -> Function of A,Y equals :Def4: id A; coherence proof A c= Y & rng id A = A by RELAT_1:71; then (Y = {} implies A = {}) & dom id A = A & rng id A c= Y by RELAT_1:71,XBOOLE_1:3; hence id A is Function of A,Y by FUNCT_2:def 1,RELSET_1:11; end; end; canceled 3; theorem for A being Subset of Y holds incl A = (id Y)|A proof let A be Subset of Y; A c= Y & incl A = id A by Def4; hence incl A = (id Y)|A by Th1; end; theorem Th54: for A being Subset of Y holds dom incl A = A & rng incl A = A proof let A be Subset of Y; incl A = id A by Def4; hence thesis by RELAT_1:71; end; theorem for A being Subset of Y st x in A holds (incl A).x = x proof let A be Subset of Y; incl A = id A by Def4; hence thesis by FUNCT_1:35; end; theorem for A being Subset of Y st x in A holds incl(A).x in Y proof let A be Subset of Y such that A1: x in A; dom incl A = A & rng incl A = A by Th54; then incl(A).x in A & A c= Y by A1,FUNCT_1:def 5; hence thesis; end; :: Projections definition let X,Y; func pr1(X,Y) -> Function means :Def5: dom it = [:X,Y:] & for x,y st x in X & y in Y holds it.[x,y] = x; existence proof deffunc F(set,set) = $1; ex f being Function st dom f = [:X,Y:] & for x,y st x in X & y in Y holds f.[x,y] = F(x,y) from Lambda_3; hence thesis; end; uniqueness proof let f1,f2 be Function such that A1: dom f1 = [:X,Y:] and A2: for x,y st x in X & y in Y holds f1.[x,y] = x and A3: dom f2 = [:X,Y:] and A4: for x,y st x in X & y in Y holds f2.[x,y] = x; for x,y st x in X & y in Y holds f1.[x,y] = f2.[x,y] proof let x,y; assume x in X & y in Y; then f1.[x,y] = x & f2.[x,y] = x by A2,A4; hence thesis; end; hence thesis by A1,A3,Th6; end; func pr2(X,Y) -> Function means :Def6: dom it = [:X,Y:] & for x,y st x in X & y in Y holds it.[x,y] = y; existence proof deffunc F(set,set) = $2; ex f being Function st dom f = [:X,Y:] & for x,y st x in X & y in Y holds f.[x,y] = F(x,y) from Lambda_3; hence thesis; end; uniqueness proof let f1,f2 be Function such that A5: dom f1 = [:X,Y:] and A6: for x,y st x in X & y in Y holds f1.[x,y] = y and A7: dom f2 = [:X,Y:] and A8: for x,y st x in X & y in Y holds f2.[x,y] = y; for x,y st x in X & y in Y holds f1.[x,y] = f2.[x,y] proof let x,y; assume x in X & y in Y; then f1.[x,y] = y & f2.[x,y] = y by A6,A8; hence thesis; end; hence thesis by A5,A7,Th6; end; end; canceled 2; theorem Th59: rng pr1(X,Y) c= X proof let x; assume x in rng pr1(X,Y); then consider p such that A1: p in dom pr1(X,Y) and A2: x = pr1(X,Y).p by FUNCT_1:def 5; p in [:X,Y:] by A1,Def5; then ex x1,y1 st x1 in X & y1 in Y & p = [x1,y1] by ZFMISC_1:def 2; hence thesis by A2,Def5; end; theorem Y <> {} implies rng pr1(X,Y) = X proof assume A1: Y <> {}; consider y being Element of Y; A2: rng pr1(X,Y) c= X by Th59; X c= rng pr1(X,Y) proof let x; assume x in X; then [x,y] in [:X,Y:] & pr1(X,Y).[x,y]=x by A1,Def5,ZFMISC_1:106; then [x,y] in dom pr1(X,Y) & pr1(X,Y).[x,y] = x by Def5; hence thesis by FUNCT_1:def 5; end; hence thesis by A2,XBOOLE_0:def 10; end; theorem Th61: rng pr2(X,Y) c= Y proof let y; assume y in rng pr2(X,Y); then consider p such that A1: p in dom pr2(X,Y) and A2: y = pr2(X,Y).p by FUNCT_1:def 5; p in [:X,Y:] by A1,Def6; then ex x1,y1 st x1 in X & y1 in Y & p = [x1,y1] by ZFMISC_1:def 2; hence thesis by A2,Def6; end; theorem X <> {} implies rng pr2(X,Y) = Y proof assume A1: X <> {}; consider x being Element of X; A2: rng pr2(X,Y) c= Y by Th61; Y c= rng pr2(X,Y) proof let y; assume y in Y; then [x,y] in [:X,Y:] & pr2(X,Y).[x,y]=y by A1,Def6,ZFMISC_1:106; then [x,y] in dom pr2(X,Y) & pr2(X,Y).[x,y] = y by Def6; hence thesis by FUNCT_1:def 5; end; hence thesis by A2,XBOOLE_0:def 10; end; definition let X,Y; redefine func pr1(X,Y) -> Function of [:X,Y:],X; coherence proof per cases; suppose A1: X = {} implies [:X,Y:] = {}; dom pr1(X,Y) = [:X,Y:] & rng pr1(X,Y) c= X by Def5,Th59; hence thesis by A1,FUNCT_2:def 1,RELSET_1:11; suppose X = {} & [:X,Y:] <> {}; hence thesis by ZFMISC_1:113; end; func pr2(X,Y) -> Function of [:X,Y:],Y; coherence proof per cases; suppose A2: Y = {} implies [:X,Y:] = {}; dom pr2(X,Y) = [:X,Y:] & rng pr2(X,Y) c= Y by Def6,Th61; hence thesis by A2,FUNCT_2:def 1,RELSET_1:11; suppose Y = {} & [:X,Y:] <> {}; hence thesis by ZFMISC_1:113; end; end; definition let X; func delta(X) -> Function means :Def7: dom it = X & for x st x in X holds it.x = [x,x]; existence proof deffunc F(set) = [$1,$1]; ex f being Function st dom f = X & for x st x in X holds f.x = F(x) from Lambda; hence thesis; end; uniqueness proof let f1,f2 be Function such that A1: dom f1 = X and A2: for x st x in X holds f1.x = [x,x] and A3: dom f2 = X and A4: for x st x in X holds f2.x = [x,x]; for x st x in X holds f1.x = f2.x proof let x; assume x in X; then f1.x = [x,x] & f2.x=[x,x] by A2,A4; hence thesis; end; hence thesis by A1,A3,FUNCT_1:9; end; end; canceled 3; theorem Th66: rng delta X c= [:X,X:] proof let y; assume y in rng delta X; then consider x such that A1: x in dom delta X and A2: y = (delta X).x by FUNCT_1:def 5; now thus x in X by A1,Def7; hence y = [x,x] by A2,Def7; end; hence thesis by ZFMISC_1:106; end; definition let X; redefine func delta(X) -> Function of X,[:X,X:]; coherence proof ( [:X,X:] = {} implies X = {} ) & dom delta X = X & rng delta X c= [:X,X:] by Def7,Th66,ZFMISC_1:113; hence thesis by FUNCT_2:def 1,RELSET_1:11; end; end; :: Complex functions definition let f,g be Function; func <:f,g:> -> Function means :Def8: dom it = dom f /\ dom g & for x st x in dom it holds it.x = [f.x,g.x]; existence proof deffunc F(set) = [f.$1,g.$1]; ex fg being Function st dom fg = dom f /\ dom g & for x st x in dom f /\ dom g holds fg.x = F(x) from Lambda; hence thesis; end; uniqueness proof let f1,f2 be Function such that A1: dom f1 = dom f /\ dom g and A2: for x st x in dom f1 holds f1.x = [f.x,g.x] and A3: dom f2 = dom f /\ dom g and A4: for x st x in dom f2 holds f2.x = [f.x,g.x]; for x st x in dom f /\ dom g holds f1.x=f2.x proof let x; assume x in dom f /\ dom g; then f1.x=[f.x,g.x] & f2.x=[f.x,g.x] by A1,A2,A3,A4; hence thesis; end; hence thesis by A1,A3,FUNCT_1:9; end; end; canceled; theorem Th68: for f,g being Function st x in dom f /\ dom g holds <:f,g:>.x = [f.x,g.x] proof let f,g be Function; assume x in dom f /\ dom g; then x in dom <:f,g:> by Def8; hence thesis by Def8; end; theorem Th69: for f,g being Function st dom f = X & dom g = X & x in X holds <:f,g:>.x = [f.x,g.x] proof let f,g be Function; assume dom f = X & dom g = X & x in X; then x in dom f /\ dom g; then x in dom <:f,g:> by Def8; hence thesis by Def8; end; theorem Th70: for f,g being Function st dom f = X & dom g = X holds dom <:f,g:> = X proof let f,g be Function; dom <:f,g:> = dom f /\ dom g by Def8; hence thesis; end; theorem Th71: for f,g being Function holds rng <:f,g:> c= [:rng f,rng g:] proof let f,g be Function; let q; assume q in rng <:f,g:>; then consider x such that A1: x in dom <:f,g:> and A2: q = <:f,g:>.x by FUNCT_1:def 5; x in dom f /\ dom g by A1,Def8; then x in dom f & x in dom g by XBOOLE_0:def 3; then f.x in rng f & g.x in rng g & q = [f.x,g.x] by A1,A2,Def8,FUNCT_1:def 5 ; hence thesis by ZFMISC_1:106; end; theorem Th72: for f,g being Function st dom f = dom g & rng f c= Y & rng g c= Z holds pr1(Y,Z)*<:f,g:> = f & pr2(Y,Z)*<:f,g:> = g proof let f,g be Function such that A1: dom f = dom g and A2: rng f c= Y & rng g c= Z; A3: [:rng f, rng g:] c= [:Y,Z:] & rng <:f,g:> c= [:rng f, rng g:] by A2,Th71,ZFMISC_1:119; dom pr1(Y,Z) = [:Y, Z:] by Def5; then rng <:f,g:> c= dom pr1(Y,Z) by A3,XBOOLE_1:1; then A4: dom(pr1(Y,Z)*<:f,g:>) = dom <:f,g:> by RELAT_1:46; then A5: dom(pr1(Y,Z)*<:f,g:>) = dom f by A1,Th70; x in dom f implies (pr1(Y,Z)*<:f,g:>).x = f.x proof assume A6: x in dom f; then A7: f.x in rng f & g.x in rng g by A1,FUNCT_1:def 5; thus (pr1(Y,Z)*<:f,g:>).x = pr1(Y,Z).(<:f,g:>.x) by A5,A6,FUNCT_1:22 .= pr1(Y,Z).[f.x,g.x] by A4,A5,A6,Def8 .= f.x by A2,A7,Def5; end; hence pr1(Y,Z)*<:f,g:> = f by A5,FUNCT_1:9; dom pr2(Y,Z) = [:Y, Z:] by Def6; then rng <:f,g:> c= dom pr2(Y,Z) by A3,XBOOLE_1:1; then A8: dom(pr2(Y,Z)*<:f,g:>) = dom <:f,g:> by RELAT_1:46; then A9: dom(pr2(Y,Z)*<:f,g:>) = dom g by A1,Th70; x in dom g implies (pr2(Y,Z)*<:f,g:>).x = g.x proof assume A10: x in dom g; then A11: f.x in rng f & g.x in rng g by A1,FUNCT_1:def 5; thus (pr2(Y,Z)*<:f,g:>).x = pr2(Y,Z).(<:f,g:>.x) by A9,A10,FUNCT_1:22 .= pr2(Y,Z).[f.x,g.x] by A8,A9,A10,Def8 .= g.x by A2,A11,Def6; end; hence pr2(Y,Z)*<:f,g:> = g by A9,FUNCT_1:9; end; theorem Th73: <:pr1(X,Y),pr2(X,Y):> = id [:X,Y:] proof dom pr1(X,Y) = [:X,Y:] & dom pr2(X,Y) = [:X,Y:] by Def5,Def6; then A1: dom <:pr1(X,Y),pr2(X,Y):> = [:X,Y:] & dom id [:X,Y:] = [:X,Y:] by Th70,RELAT_1:71; for x,y st x in X & y in Y holds <:pr1(X,Y),pr2(X,Y):>.[x,y] = (id [:X,Y:]).[x,y] proof let x,y; assume A2: x in X & y in Y; then A3: [x,y] in [:X,Y:] by ZFMISC_1:106; hence <:pr1(X,Y),pr2(X,Y):>.[x,y] = [pr1(X,Y).[x,y],pr2(X,Y).[x,y]] by A1,Def8 .= [x,pr2(X,Y).[x,y]] by A2,Def5 .= [x,y] by A2,Def6 .= (id [:X,Y:]).[x,y] by A3,FUNCT_1:35; end; hence thesis by A1,Th6; end; theorem Th74: for f,g,h,k being Function st dom f = dom g & dom k = dom h & <:f,g:> = <:k,h:> holds f = k & g = h proof let f,g,h,k be Function such that A1: dom f = dom g and A2: dom k = dom h and A3: <:f,g:> = <:k,h:>; A4: dom <:f,g:> = dom f & dom <:k,h:> = dom k by A1,A2,Th70; x in dom f implies f.x = k.x proof assume x in dom f; then <:f,g:>.x = [f.x,g.x] & <:k,h:>.x = [k.x,h.x] by A3,A4,Def8; hence thesis by A3,ZFMISC_1:33; end; hence f = k by A3,A4,FUNCT_1:9; A5: dom <:f,g:> = dom g & dom <:k,h:> = dom h by A1,A2,Th70; x in dom g implies g.x = h.x proof assume x in dom g; then <:f,g:>.x = [f.x,g.x] & <:k,h:>.x = [k.x,h.x] by A3,A5,Def8; hence thesis by A3,ZFMISC_1:33; end; hence thesis by A3,A5,FUNCT_1:9; end; theorem for f,g,h being Function holds <:f*h,g*h:> = <:f,g:>*h proof let f,g,h be Function; x in dom <:f*h,g*h:> iff x in dom(<:f,g:>*h) proof thus x in dom <:f*h,g*h:> implies x in dom(<:f,g:>*h) proof assume x in dom <:f*h,g*h:>; then x in dom(f*h) /\ dom(g*h) by Def8; then A1: x in dom(f*h) & x in dom(g*h) by XBOOLE_0:def 3; then h.x in dom f & h.x in dom g by FUNCT_1:21; then h.x in dom f /\ dom g by XBOOLE_0:def 3; then h.x in dom <:f,g:> & x in dom h by A1,Def8,FUNCT_1:21; hence x in dom(<:f,g:>*h) by FUNCT_1:21; end; assume x in dom(<:f,g:>*h); then h.x in dom <:f,g:> & x in dom h by FUNCT_1:21; then h.x in dom f /\ dom g & x in dom h by Def8; then h.x in dom f & h.x in dom g & x in dom h by XBOOLE_0:def 3; then x in dom(f*h) & x in dom(g*h) by FUNCT_1:21; then x in dom(f*h) /\ dom(g*h) by XBOOLE_0:def 3; hence thesis by Def8; end; then A2: dom <:f*h,g*h:> = dom(<:f,g:>*h) by TARSKI:2; for x st x in dom <:f*h,g*h:> holds <:f*h,g*h:>.x = (<:f,g:>*h).x proof let x; assume A3: x in dom <:f*h,g*h:>; then x in dom(f*h) /\ dom(g*h) by Def8; then A4: x in dom(f*h) & x in dom(g*h) by XBOOLE_0:def 3; then h.x in dom f & h.x in dom g by FUNCT_1:21; then A5: h.x in dom f /\ dom g by XBOOLE_0:def 3; then A6: h.x in dom <:f,g:> & x in dom h by A4,Def8,FUNCT_1:21; thus <:f*h,g*h:>.x = [(f*h).x,(g*h).x] by A3,Def8 .= [f.(h.x),(g*h).x] by A4,FUNCT_1:22 .= [f.(h.x),g.(h.x)] by A4,FUNCT_1:22 .= <:f,g:>.(h.x) by A5,Th68 .= (<:f,g:>*h).x by A6,FUNCT_1:23; end; hence thesis by A2,FUNCT_1:9; end; theorem for f,g being Function holds <:f,g:>.:A c= [:f.:A,g.:A:] proof let f,g be Function; let y; assume y in <:f,g:>.:A; then consider x such that A1: x in dom <:f,g:> and A2: x in A and A3: y = <:f,g:>.x by FUNCT_1:def 12; x in dom f /\ dom g by A1,Def8; then x in dom f & x in dom g by XBOOLE_0:def 3; then f.x in f.:A & g.x in g.: A & y = [f.x,g.x] by A1,A2,A3,Def8,FUNCT_1:def 12; hence y in [:f.:A,g.:A:] by ZFMISC_1:def 2; end; theorem for f,g being Function holds <:f,g:>"[:B,C:] = f"B /\ g"C proof let f,g be Function; x in <:f,g:>"[:B,C:] iff x in f"B /\ g"C proof thus x in <:f,g:>"[:B,C:] implies x in f"B /\ g"C proof assume A1: x in <:f,g:>"[:B,C:]; then <:f,g:>.x in [:B,C:] by FUNCT_1:def 13; then consider y1,y2 such that A2: y1 in B & y2 in C and A3: <:f,g:>.x = [y1,y2] by ZFMISC_1:def 2; x in dom <:f,g:> by A1,FUNCT_1:def 13; then x in dom f /\ dom g & [y1,y2] = [f.x,g.x] by A3,Def8; then x in dom f & x in dom g & y1 =f.x & y2 = g.x by XBOOLE_0:def 3, ZFMISC_1:33; then x in f"B & x in g"C by A2,FUNCT_1:def 13; hence x in f"B /\ g"C by XBOOLE_0:def 3; end; assume x in f"B /\ g"C; then x in f"B & x in g"C by XBOOLE_0:def 3; then x in dom f & x in dom g & f.x in B & g.x in C by FUNCT_1:def 13; then x in dom f /\ dom g & [f.x,g.x] in [:B,C:] by XBOOLE_0:def 3, ZFMISC_1:def 2; then x in dom <:f,g:> & <:f,g:>.x in [:B,C:] by Def8,Th68; hence thesis by FUNCT_1:def 13; end; hence thesis by TARSKI:2; end; theorem Th78: for f being Function of X,Y for g being Function of X,Z st (Y = {} implies X = {}) & (Z = {} implies X = {}) holds <:f,g:> is Function of X,[:Y,Z:] proof let f be Function of X,Y; let g be Function of X,Z; assume A1: (Y = {} implies X = {}) & (Z = {} implies X = {}); per cases; suppose A2: [:Y,Z:] = {} implies X = {}; rng f c= Y & rng g c= Z by RELSET_1:12; then [:rng f,rng g:] c= [:Y,Z:] & rng <:f,g:> c= [:rng f,rng g:] & dom f = X & dom g = X by A1,Th71,FUNCT_2:def 1,ZFMISC_1:119; then dom<:f,g:> = X & rng<:f,g:> c= [:Y,Z:] by Th70,XBOOLE_1:1; hence thesis by A2,FUNCT_2:def 1,RELSET_1:11; suppose [:Y,Z:] = {} & X <> {}; hence thesis by A1,ZFMISC_1:113; end; definition let X,D1,D2; let f1 be Function of X,D1; let f2 be Function of X,D2; redefine func <:f1,f2:> -> Function of X,[:D1,D2:]; coherence by Th78; end; theorem for f1 being Function of C,D1 for f2 being Function of C,D2 for c being Element of C holds <:f1,f2:>.c = [f1.c,f2.c] proof let f1 be Function of C,D1; let f2 be Function of C,D2; let c be Element of C; c in C & dom f1 = C & dom f2 = C by FUNCT_2:def 1; hence thesis by Th69; end; theorem for f being Function of X,Y for g being Function of X,Z holds rng <:f,g:> c= [:Y,Z:] proof let f be Function of X,Y; let g be Function of X,Z; rng f c= Y & rng g c= Z by RELSET_1:12; then [:rng f,rng g:] c= [:Y,Z:] & rng <:f,g:> c= [:rng f,rng g:] by Th71,ZFMISC_1:119; hence thesis by XBOOLE_1:1; end; theorem Th81: for f being Function of X,Y for g being Function of X,Z st (Y = {} implies X = {}) & (Z = {} implies X = {}) holds pr1(Y,Z)*<:f,g:> = f & pr2(Y,Z)*<:f,g:> = g proof let f be Function of X,Y; let g be Function of X,Z; assume (Y = {} implies X = {}) & (Z = {} implies X = {}); then dom f = X & dom g = X & rng f c= Y & rng g c= Z by FUNCT_2:def 1,RELSET_1:12; hence thesis by Th72; end; theorem for f being Function of X,D1 for g being Function of X,D2 holds pr1(D1,D2)*<:f,g:> = f & pr2(D1,D2)*<:f,g:> = g by Th81; theorem for f1,f2 being Function of X,Y for g1,g2 being Function of X,Z st (Y = {} implies X = {}) & (Z = {} implies X = {}) & <:f1,g1:> = <:f2,g2:> holds f1 = f2 & g1 = g2 proof let f1,f2 be Function of X,Y; let g1,g2 be Function of X,Z; assume (Y = {} implies X = {}) & (Z = {} implies X = {}); then dom f1 = X & dom f2 = X & dom g1 = X & dom g2 = X by FUNCT_2:def 1; hence thesis by Th74; end; theorem for f1,f2 being Function of X,D1 for g1,g2 being Function of X,D2 st <:f1,g1:> = <:f2,g2:> holds f1 = f2 & g1 = g2 proof let f1,f2 be Function of X,D1; let g1,g2 be Function of X,D2; dom f1 = X & dom f2 = X & dom g1 = X & dom g2 = X by FUNCT_2:def 1; hence thesis by Th74; end; :: Product-functions definition let f,g be Function; func [:f,g:] -> Function means :Def9: dom it = [:dom f, dom g:] & for x,y st x in dom f & y in dom g holds it.[x,y] = [f.x,g.y]; existence proof deffunc F(set,set) = [f.$1,g.$2]; ex F being Function st dom F = [:dom f,dom g:] & for x,y st x in dom f & y in dom g holds F.[x,y] = F(x,y) from Lambda_3; hence thesis; end; uniqueness proof let fg1, fg2 be Function such that A1: dom fg1 = [:dom f, dom g:] and A2: for x,y st x in dom f & y in dom g holds fg1.[x,y] = [f.x,g.y] and A3: dom fg2 = [:dom f, dom g:] and A4: for x,y st x in dom f & y in dom g holds fg2.[x,y] = [f.x,g.y]; for p st p in [:dom f,dom g:] holds fg1.p=fg2.p proof let p; assume p in [:dom f,dom g:]; then consider x,y such that A5: x in dom f & y in dom g & p = [x,y] by ZFMISC_1:def 2; fg1.p = [f.x,g.y] & fg2.p = [f.x,g.y] by A2,A4,A5; hence thesis; end; hence thesis by A1,A3,FUNCT_1:9; end; end; canceled; theorem Th86: for f,g being Function, x,y st [x,y] in [:dom f,dom g:] holds [:f,g:].[x,y] = [f.x,g.y] proof let f,g be Function, x,y; assume [x,y] in [:dom f,dom g:]; then x in dom f & y in dom g by ZFMISC_1:106; hence thesis by Def9; end; theorem Th87: for f,g being Function holds [:f,g:] = <:f*pr1(dom f,dom g),g*pr2(dom f,dom g):> proof let f,g be Function; A1: rng pr1(dom f,dom g) c= dom f & rng pr2(dom f,dom g) c= dom g by Th59,Th61; A2: dom pr1(dom f,dom g) = [:dom f,dom g:] & dom pr2(dom f,dom g) = [:dom f,dom g:] by Def5,Def6; then dom(f*pr1(dom f,dom g)) = [:dom f,dom g:] & dom(g*pr2(dom f,dom g)) = [:dom f,dom g:] by A1,RELAT_1:46; then A3: dom [:f,g:] = [:dom f,dom g:] & dom <:f*pr1(dom f,dom g),g*pr2(dom f,dom g):> = [:dom f,dom g:] by Def9,Th70; for x,y st x in dom f & y in dom g holds [:f,g:].[x,y] = <:f*pr1(dom f,dom g),g*pr2(dom f,dom g):>.[x,y] proof let x,y; assume A4: x in dom f & y in dom g; then A5: [x,y] in [:dom f,dom g:] by ZFMISC_1:106; thus [:f,g:].[x,y] = [f.x,g.y] by A4,Def9 .= [f.(pr1(dom f,dom g).[x,y]),g.y] by A4,Def5 .= [f.(pr1(dom f,dom g).[x,y]),g.(pr2(dom f,dom g).[x,y])] by A4,Def6 .= [(f*pr1(dom f,dom g)).[x,y],g.(pr2(dom f,dom g).[x,y])] by A2,A5,FUNCT_1:23 .= [(f*pr1(dom f,dom g)).[x,y],(g*pr2(dom f,dom g)).[x,y]] by A2,A5,FUNCT_1:23 .= <:f*pr1(dom f,dom g),g*pr2(dom f,dom g):>.[x,y] by A3,A5,Def8; end; hence thesis by A3,Th6; end; theorem Th88: for f,g being Function holds rng [:f,g:] = [:rng f,rng g:] proof let f,g be Function; q in rng [:f,g:] iff q in [:rng f,rng g:] proof thus q in rng [:f,g:] implies q in [:rng f,rng g:] proof assume q in rng [:f,g:]; then consider p such that A1: p in dom [:f,g:] and A2: q = [:f,g:].p by FUNCT_1:def 5; p in [:dom f, dom g:] by A1,Def9; then consider x,y such that A3: x in dom f & y in dom g and A4: p = [x,y] by ZFMISC_1:def 2; f.x in rng f & g.y in rng g & q = [f.x,g.y] by A2,A3,A4,Def9,FUNCT_1: def 5; hence thesis by ZFMISC_1:106; end; assume q in [:rng f,rng g:]; then consider y1,y2 such that A5: y1 in rng f and A6: y2 in rng g and A7: q = [y1,y2] by ZFMISC_1:def 2; consider x1 such that A8: x1 in dom f and A9: y1 = f.x1 by A5,FUNCT_1:def 5; consider x2 such that A10: x2 in dom g and A11: y2 = g.x2 by A6,FUNCT_1:def 5; [x1,x2] in [:dom f,dom g:] & [:f,g:].[x1,x2]=q & dom [:f,g:]=[:dom f,dom g:] by A7,A8,A9,A10,A11,Def9,ZFMISC_1:106; hence thesis by FUNCT_1:def 5; end; hence thesis by TARSKI:2; end; theorem Th89: for f,g being Function st dom f = X & dom g = X holds <:f,g:> = [:f,g:]*(delta X) proof let f,g be Function such that A1: dom f = X and A2: dom g = X; rng delta X c= [:X,X:] by Th66; then rng delta X c= dom [:f,g:] & dom delta X = X & dom f /\ dom g = X by A1,A2,Def7,Def9; then A3: dom <:f,g:> = X & dom([:f,g:]*(delta X)) = X by Def8,RELAT_1:46; for x st x in X holds <:f,g:>.x = ([:f,g:]*(delta X)).x proof let x; assume A4: x in X; hence <:f,g:>.x = [f.x,g.x] by A3,Def8 .= [:f,g:].[x,x] by A1,A2,A4,Def9 .= [:f,g:].((delta X).x) by A4,Def7 .= ([:f,g:]*(delta X)).x by A3,A4,FUNCT_1:22; end; hence thesis by A3,FUNCT_1:9; end; theorem [:id X, id Y:] = id [:X,Y:] proof A1: dom id X = X & dom id Y = Y by RELAT_1:71; rng pr1(X,Y) c= X & rng pr2(X,Y) c= Y by Th59,Th61; then (id X)*pr1(X ,Y) = pr1(X,Y) & (id Y)*pr2(X,Y) = pr2(X,Y) by RELAT_1:79; hence [:id X, id Y:] = <:pr1(X ,Y),pr2(X,Y):> by A1,Th87 .= id [:X,Y:] by Th73; end; theorem for f,g,h,k being Function holds [:f,h:]*<:g,k:> = <:f*g,h*k:> proof let f,g,h,k be Function; x in dom([:f,h:]*<:g,k:>) iff x in dom <:f*g,h*k:> proof thus x in dom([:f,h:]*<:g,k:>) implies x in dom <:f*g,h*k:> proof assume x in dom([:f,h:]*<:g,k:>); then x in dom <:g,k:> & <:g,k:>.x in dom [:f,h:] by FUNCT_1:21; then x in dom <:g,k:> & [g.x,k.x] in dom [:f,h:] by Def8; then [g.x,k.x] in [:dom f,dom h:] & x in dom g /\ dom k by Def8,Def9; then g.x in dom f & k.x in dom h & x in dom g & x in dom k by XBOOLE_0:def 3,ZFMISC_1:106; then x in dom(f*g) & x in dom(h*k) by FUNCT_1:21; then x in dom(f*g) /\ dom(h*k) by XBOOLE_0:def 3; hence thesis by Def8; end; assume x in dom <:f*g,h*k:>; then x in dom(f*g) /\ dom(h*k) by Def8; then x in dom(f*g) & x in dom(h*k) by XBOOLE_0:def 3; then g.x in dom f & k.x in dom h & x in dom g & x in dom k by FUNCT_1:21; then [g.x,k.x] in [:dom f,dom h:] & x in dom g /\ dom k by XBOOLE_0:def 3,ZFMISC_1:106; then [g.x,k.x] in dom [:f,h:] & x in dom <:g,k:> by Def8,Def9; then <:g,k:>.x in dom [:f,h:] & x in dom <:g,k:> by Def8; hence thesis by FUNCT_1:21; end; then A1: dom([:f,h:]*<:g,k:>) = dom <:f*g,h*k:> by TARSKI:2; for x st x in dom([:f,h:]*<:g,k:>) holds ([:f,h:]*<:g,k:>).x = <:f*g,h*k:> . x proof let x; assume A2: x in dom([:f,h:]*<:g,k:>); then A3: x in dom <:g,k:> & <:g,k:>.x in dom [:f,h:] by FUNCT_1:21; then [g.x,k.x] in dom [:f,h:] by Def8; then [g.x,k.x] in [:dom f,dom h:] & x in dom g /\ dom k by A3,Def8,Def9; then A4: g.x in dom f & k.x in dom h & x in dom g & x in dom k by XBOOLE_0:def 3,ZFMISC_1:106; then x in dom(f*g) & x in dom(h*k) by FUNCT_1:21; then A5: x in dom(f*g) /\ dom(h*k) by XBOOLE_0:def 3; thus ([:f,h:]*<:g,k:>).x = [:f,h:].(<:g,k:>.x) by A2,FUNCT_1:22 .= [:f,h:].[g.x,k.x] by A3,Def8 .= [f.(g.x),h.(k.x)] by A4,Def9 .= [(f*g).x,h.(k.x)] by A4,FUNCT_1:23 .= [(f*g).x,(h*k).x] by A4,FUNCT_1:23 .= <:f*g,h*k:>.x by A5,Th68; end; hence thesis by A1,FUNCT_1:9; end; theorem for f,g,h,k being Function holds [:f,h:]*[:g,k:] = [:f*g,h*k:] proof let f,g,h,k be Function; p in dom([:f,h:]*[:g,k:]) iff p in [:dom(f*g),dom(h*k):] proof A1: dom [:f,h:] = [:dom f, dom h:] by Def9; A2: dom [:g,k:] = [:dom g, dom k:] by Def9; thus p in dom([:f,h:]*[:g,k:]) implies p in [:dom(f*g),dom(h*k):] proof assume p in dom([:f,h:]*[:g,k:]); then A3: p in dom [:g,k:] & [:g,k:].p in dom [:f,h:] by FUNCT_1:21; then consider x1,x2 such that A4: x1 in dom g & x2 in dom k and A5: p = [x1,x2] by A2,ZFMISC_1:def 2; [g.x1,k.x2] in dom [:f,h:] by A2,A3,A5,Th86; then g.x1 in dom f & k.x2 in dom h by A1,ZFMISC_1:106; then x1 in dom(f*g) & x2 in dom(h*k) by A4,FUNCT_1:21; hence thesis by A5,ZFMISC_1:106; end; assume p in [:dom(f*g),dom(h*k):]; then consider x1,x2 such that A6: x1 in dom(f*g) & x2 in dom(h*k) and A7: p = [x1,x2] by ZFMISC_1:def 2; g.x1 in dom f & k.x2 in dom h & x1 in dom g & x2 in dom k by A6,FUNCT_1:21; then [g.x1,k.x2] in dom [:f,h:] & [x1,x2] in dom [:g,k:] by A1,A2,ZFMISC_1:106; then [:g,k:].[x1,x2] in dom [:f,h:] & [x1,x2] in dom [:g,k:] by A2,Th86 ; hence thesis by A7,FUNCT_1:21; end; then A8: dom([:f,h:]*[:g,k:]) = [:dom(f*g),dom(h*k):] & [:dom(f*g),dom(h*k):] = dom [:f*g,h*k:] by Def9,TARSKI:2; for x,y st x in dom(f*g) & y in dom(h*k) holds ([:f,h:]*[:g,k:]).[x,y] = [:f*g,h*k:].[x,y] proof let x,y such that A9: x in dom(f*g) and A10: y in dom(h*k); A11: g.x in dom f & k.y in dom h by A9,A10,FUNCT_1:21; A12: x in dom g & y in dom k by A9,A10,FUNCT_1:21; then [x,y] in [:dom g, dom k:] & [g.x,k.y] in [:dom f,dom h:] by A11,ZFMISC_1:106; then [x,y] in [:dom g, dom k:] & [:g,k:].[x,y] in [:dom f,dom h:] by Th86 ; then [x,y] in dom [:g,k:] & [:g,k:].[x,y] in dom [:f,h:] by Def9; hence ([:f,h:]*[:g,k:]).[x,y] = [:f,h:].([:g,k:].[x,y]) by FUNCT_1:23 .= [:f,h:].[g.x,k.y] by A12,Def9 .= [f.(g.x),h.(k.y)] by A11,Def9 .= [(f*g).x,h.(k.y)] by A9,FUNCT_1:22 .= [(f*g).x,(h*k).y] by A10,FUNCT_1:22 .= [:f*g,h*k:].[x,y] by A9,A10,Def9; end; hence thesis by A8,Th6; end; theorem for f,g being Function holds [:f,g:].:[:B,A:] = [:f.:B,g.:A:] proof let f,g be Function; q in [:f,g:].:[:B,A:] iff q in [:f.:B,g.:A:] proof thus q in [:f,g:].:[:B,A:] implies q in [:f.:B,g.:A:] proof assume q in [:f,g:].:[:B,A:]; then consider p such that A1: p in dom [:f,g:] and A2: p in [:B,A:] and A3: q = [:f,g:].p by FUNCT_1:def 12; dom [:f,g:] = [:dom f,dom g:] by Def9; then consider x,y such that A4: x in dom f & y in dom g and A5: p =[x,y] by A1,ZFMISC_1:def 2; x in B & y in A by A2,A5,ZFMISC_1:106; then f.x in f.:B & g.y in g.:A & q=[f.x,g.y] by A3,A4,A5,Def9,FUNCT_1:def 12; hence q in [:f.:B,g.:A:] by ZFMISC_1:106; end; assume q in [:f.:B,g.:A:]; then consider y1,y2 such that A6: y1 in f.:B and A7: y2 in g.:A and A8: q = [y1,y2] by ZFMISC_1:def 2; consider x1 such that A9: x1 in dom f & x1 in B and A10: y1 = f.x1 by A6,FUNCT_1:def 12; consider x2 such that A11: x2 in dom g & x2 in A and A12: y2 = g.x2 by A7,FUNCT_1:def 12; [x1,x2] in [:dom f,dom g:] & [:dom f,dom g:] = dom [:f,g:] & [x1,x2] in [:B,A:] & [:f,g:].[x1,x2] = [f.x1,g.x2] by A9,A11,Def9,ZFMISC_1:106; hence thesis by A8,A10,A12,FUNCT_1:def 12; end; hence thesis by TARSKI:2; end; theorem for f,g being Function holds [:f,g:]"[:B,A:] = [:f"B,g"A:] proof let f,g be Function; q in [:f,g:]"[:B,A:] iff q in [:f"B,g"A:] proof thus q in [:f,g:]"[:B,A:] implies q in [:f"B,g"A:] proof assume A1: q in [:f,g:]"[:B,A:]; then q in dom [:f,g:] by FUNCT_1:def 13; then q in [:dom f,dom g:] by Def9; then consider x1,x2 such that A2: x1 in dom f & x2 in dom g and A3: q = [x1,x2] by ZFMISC_1:def 2; [:f,g:].q in [:B,A:] by A1,FUNCT_1:def 13; then [f.x1,g.x2] in [:B,A:] by A2,A3,Def9; then f.x1 in B & g.x2 in A by ZFMISC_1:106; then x1 in f"B & x2 in g"A by A2,FUNCT_1:def 13; hence thesis by A3,ZFMISC_1:106; end; assume q in [:f"B,g"A:]; then consider x1,x2 such that A4: x1 in f"B and A5: x2 in g"A and A6: q = [x1,x2] by ZFMISC_1:def 2; x1 in dom f & x2 in dom g & f.x1 in B & g.x2 in A by A4,A5,FUNCT_1:def 13 ; then [x1,x2] in [:dom f,dom g:] & [:dom f,dom g:] = dom [:f,g:] & [:f,g:].[x1,x2] = [f.x1,g.x2] & [f.x1,g.x2] in [:B,A:] by Def9,ZFMISC_1:106; hence thesis by A6,FUNCT_1:def 13; end; hence thesis by TARSKI:2; end; theorem Th95: for f being Function of X,Y for g being Function of V,Z holds [:f,g:] is Function of [:X,V:],[:Y,Z:] proof let f be Function of X,Y; let g be Function of V,Z; per cases; suppose A1: [:Y,Z:] = {} implies [:X,V:] = {}; now per cases by A1; suppose [:Y,Z:] <> {}; then A2: (Y = {} implies X = {}) & (Z = {} implies V = {}) by ZFMISC_1:113; rng f c= Y & rng g c= Z by RELSET_1:12; then [:rng f,rng g:] c= [:Y,Z:] & rng [:f,g:] = [:rng f,rng g:] by Th88,ZFMISC_1:119; then dom f = X & dom g = V & rng [:f,g:] c= [:Y,Z:] by A2,FUNCT_2:def 1; then dom[:f,g:] = [:X,V:] & rng [:f,g:] c= [:Y,Z:] by Def9; hence thesis by A1,FUNCT_2:def 1,RELSET_1:11; suppose A3: [:X,V:] = {}; then X = {} or V = {} by ZFMISC_1:113; then dom f = {} or dom g = {} by FUNCT_2:def 1; then [:dom f,dom g:] = {} by ZFMISC_1:113; then A4: dom[:f,g:] = [:X,V:] by A3,Def9; rng f c= Y & rng g c= Z by RELSET_1:12; then [:rng f,rng g:] c= [:Y,Z:] by ZFMISC_1:119; then rng[:f,g:] c= [:Y,Z:] by Th88; hence thesis by A1,A4,FUNCT_2:def 1,RELSET_1:11; end; hence thesis; suppose A5: [:Y,Z:] = {} & [:X,V:] <> {}; then (Y = {} or Z = {}) & X <> {} & V <> {} by ZFMISC_1:113; then f = {} or g = {} by FUNCT_2:def 1; then [:dom f,dom g:] = {} by RELAT_1:60,ZFMISC_1:113; then A6: dom [:f,g:] = {} by Def9; then A7: [:f,g:] = {} by RELAT_1:64; rng[:f,g:] = {} by A6,RELAT_1:65; then dom [:f,g:] c= [:X,V:] & rng[:f,g:] c= [:Y,Z:] by A6,XBOOLE_1:2; then reconsider R = [:f,g:] as Relation of [:X,V:],[:Y,Z:] by RELSET_1:11; R is quasi_total by A5,A7,FUNCT_2:def 1; hence thesis; end; definition let X1,X2,Y1,Y2; let f1 be Function of X1,Y1; let f2 be Function of X2,Y2; redefine func [:f1,f2:] -> Function of [:X1,X2:],[:Y1,Y2:]; coherence by Th95; end; theorem for f1 being Function of C1,D1 for f2 being Function of C2,D2 for c1 being Element of C1 for c2 being Element of C2 holds [:f1,f2:].[c1,c2] = [f1.c1,f2.c2] proof let f1 be Function of C1,D1; let f2 be Function of C2,D2; let c1 be Element of C1; let c2 be Element of C2; c1 in C1 & c2 in C2 & dom f1 = C1 & dom f2 = C2 by FUNCT_2:def 1; hence thesis by Def9; end; theorem for f1 being Function of X1,Y1 for f2 being Function of X2,Y2 st (Y1 = {} implies X1 = {}) & (Y2 = {} implies X2 = {}) holds [:f1,f2:] = <:f1*pr1(X1,X2),f2*pr2(X1,X2):> proof let f1 be Function of X1,Y1; let f2 be Function of X2,Y2; assume (Y1 = {} implies X1 = {}) & (Y2 = {} implies X2 = {}); then dom f1 = X1 & dom f2 = X2 by FUNCT_2:def 1; hence thesis by Th87; end; theorem for f1 being Function of X1,D1 for f2 being Function of X2,D2 holds [:f1,f2:] = <:f1*pr1(X1,X2),f2*pr2(X1,X2):> proof let f1 be Function of X1,D1; let f2 be Function of X2,D2; dom f1 = X1 & dom f2 = X2 by FUNCT_2:def 1; hence thesis by Th87; end; theorem for f1 being Function of X,Y1 for f2 being Function of X,Y2 st (Y1 = {} implies X = {}) & (Y2 = {} implies X = {}) holds <:f1,f2:> = [:f1,f2:]*(delta X) proof let f1 be Function of X,Y1; let f2 be Function of X,Y2; assume (Y1 = {} implies X = {}) & (Y2 = {} implies X = {}); then dom f1 = X & dom f2 = X by FUNCT_2:def 1; hence thesis by Th89; end; theorem for f1 being Function of X,D1 for f2 being Function of X,D2 holds <:f1,f2:> = [:f1,f2:]*(delta X) proof let f1 be Function of X,D1; let f2 be Function of X,D2; dom f1 = X & dom f2 = X by FUNCT_2:def 1; hence thesis by Th89; end;

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