:: Functions from a Set to a Set
:: by Czes{\l}aw Byli\'nski
::
:: Received April 6, 1989
:: Copyright (c) 1990-2016 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies RELAT_1, XBOOLE_0, PARTFUN1, FUNCT_1, TARSKI, SUBSET_1, ZFMISC_1,
RELAT_2, MCART_1, SETFAM_1, FUNCT_2, GROUP_9;
notations TARSKI, XBOOLE_0, ZFMISC_1, XTUPLE_0, SUBSET_1, SETFAM_1, MCART_1,
RELAT_1, RELAT_2, FUNCT_1, RELSET_1, PARTFUN1;
constructors RELAT_2, PARTFUN1, MCART_1, SETFAM_1, RELSET_1, XTUPLE_0;
registrations XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, PARTFUN1, RELSET_1,
GRFUNC_1, ZFMISC_1;
requirements SUBSET, BOOLE;
begin :: Function from a set into a set
reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
definition
let X,Y;
let R be Relation of X,Y;
attr R is quasi_total means
:: FUNCT_2:def 1
X = dom R if Y <> {} otherwise R = {};
end;
registration
let X,Y;
cluster quasi_total for PartFunc of X,Y;
end;
registration
let X,Y;
cluster total -> quasi_total for Relation of X,Y;
end;
definition
let X,Y;
mode Function of X,Y is quasi_total PartFunc of X,Y;
end;
registration let X be empty set, Y be set;
cluster quasi_total -> total for Relation of X,Y;
end;
registration let X be set, Y be non empty set;
cluster quasi_total -> total for Relation of X,Y;
end;
registration let X be set;
cluster quasi_total -> total for Relation of X,X;
end;
registration let X be set;
cluster quasi_total -> total for Relation of [:X,X:],X;
end;
theorem :: FUNCT_2:1
for f being Function holds f is Function of dom f, rng f;
theorem :: FUNCT_2:2
for f being Function st rng f c= Y holds f is Function of dom f, Y;
theorem :: FUNCT_2:3
for f being Function st dom f = X & for x st x in X holds f.x in Y
holds f is Function of X,Y;
theorem :: FUNCT_2:4
for f being Function of X,Y st Y <> {} & x in X holds f.x in rng f;
theorem :: FUNCT_2:5
for f being Function of X,Y st Y <> {} & x in X holds f.x in Y;
theorem :: FUNCT_2:6
for f being Function of X,Y st (Y = {} implies X = {}) & rng f c= Z
holds f is Function of X,Z;
theorem :: FUNCT_2:7
for f being Function of X,Y st (Y = {} implies X = {}) & Y c= Z holds
f is Function of X,Z;
scheme :: FUNCT_2:sch 1
FuncEx1{X, Y() -> set, P[object,object]}:
ex f being Function of X(),Y() st
for x being object st x in X() holds P[x,f.x]
provided
for x being object st x in X() ex y being object st y in Y() & P[x,y];
scheme :: FUNCT_2:sch 2
Lambda1{X, Y() -> set, F(object)->object}:
ex f being Function of X(),Y() st
for x being object st x in X() holds f.x = F(x)
provided
for x being object st x in X() holds F(x) in Y();
definition
let X,Y;
func Funcs(X,Y) -> set means
:: FUNCT_2:def 2
x in it iff ex f being Function st x = f & dom f = X & rng f c= Y;
end;
theorem :: FUNCT_2:8
for f being Function of X,Y st Y = {} implies X = {} holds f in Funcs(X,Y);
theorem :: FUNCT_2:9
for f being Function of X,X holds f in Funcs(X,X);
registration
let X be set, Y be non empty set;
cluster Funcs(X,Y) -> non empty;
end;
registration
let X be set;
cluster Funcs(X,X) -> non empty;
end;
registration
let X be non empty set, Y be empty set;
cluster Funcs(X,Y) -> empty;
end;
theorem :: FUNCT_2:10
for f being Function of X,Y st for y being object st y in Y
ex x being object st x in X & y = f.x
holds rng f = Y;
theorem :: FUNCT_2:11
for f being Function of X,Y st y in rng f
ex x being object st x in X & f.x = y;
theorem :: FUNCT_2:12
for f1,f2 being Function of X,Y st
for x being object st x in X holds f1.x = f2.x
holds f1 = f2;
theorem :: FUNCT_2:13
for f being quasi_total Relation of X,Y for g being quasi_total
Relation of Y,Z st Y = {} implies Z = {} or X = {} holds f*g is quasi_total;
theorem :: FUNCT_2:14
for f being Function of X,Y for g being Function of Y,Z st Z <> {} &
rng f = Y & rng g = Z holds rng(g*f) = Z;
theorem :: FUNCT_2:15
for x being object
for f being Function of X,Y, g being Function st Y <> {} & x in
X holds (g*f).x = g.(f.x);
theorem :: FUNCT_2:16
for f being Function of X,Y st Y <> {} holds rng f = Y iff for Z st Z
<> {} for g,h being Function of Y,Z st g*f = h*f holds g = h;
theorem :: FUNCT_2:17
for f being Relation of X,Y holds (id X)*f = f & f*(id Y) = f;
theorem :: FUNCT_2:18
for f being Function of X,Y for g being Function of Y,X st f*g = id Y
holds rng f = Y;
theorem :: FUNCT_2:19
for f being Function of X,Y st Y = {} implies X = {} holds f is
one-to-one iff
for x1,x2 being object st x1 in X & x2 in X & f.x1 = f.x2 holds x1 = x2;
theorem :: FUNCT_2:20
for f being Function of X,Y for g being Function of Y,Z st (Z = {}
implies Y = {}) & g*f is one-to-one holds f is one-to-one;
theorem :: FUNCT_2:21
for f being Function of X,Y st X <> {} & Y <> {} holds f is one-to-one
iff for Z for g,h being Function of Z,X st f*g = f*h holds g = h;
theorem :: FUNCT_2:22
for f being Function of X,Y for g being Function of Y,Z st Z <> {} &
rng(g*f) = Z & g is one-to-one holds rng f = Y;
definition
let Y be set;
let f be Y-valued Relation;
attr f is onto means
:: FUNCT_2:def 3
rng f = Y;
end;
theorem :: FUNCT_2:23
for f being Function of X,Y for g being Function of Y,X st g*f = id X
holds f is one-to-one & g is onto;
theorem :: FUNCT_2:24
for f being Function of X,Y for g being Function of Y,Z st (Z = {}
implies Y = {}) & g*f is one-to-one & rng f = Y holds f is one-to-one & g is
one-to-one;
theorem :: FUNCT_2:25
for f being Function of X,Y st f is one-to-one & rng f = Y holds
f" is Function of Y,X;
theorem :: FUNCT_2:26
for f being Function of X,Y st Y <> {} & f is one-to-one & x in X
holds (f").(f.x) = x;
theorem :: FUNCT_2:27
for X be set, Y,Z be non empty set for f be Function of X,Y
for g be Function of Y,Z holds f is onto & g is onto implies g*f is onto;
theorem :: FUNCT_2:28
for f being Function of X,Y for g being Function of Y,X st X <> {} & Y
<> {} & rng f = Y & f is one-to-one &
for y,x being object holds y in Y & g.y = x iff x in X
& f.x = y holds g = f";
theorem :: FUNCT_2:29
for f being Function of X,Y st Y <> {} & rng f = Y & f is one-to-one
holds f"*f = id X & f*f" = id Y;
theorem :: FUNCT_2:30
for f being Function of X,Y for g being Function of Y,X st X <> {} & Y
<> {} & rng f = Y & g*f = id X & f is one-to-one holds g = f";
theorem :: FUNCT_2:31
for f being Function of X,Y st Y <> {} & ex g being Function of Y,X st
g*f = id X holds f is one-to-one;
theorem :: FUNCT_2:32
for f being Function of X,Y st (Y = {} implies X = {}) & Z c= X
holds f|Z is Function of Z,Y;
theorem :: FUNCT_2:33
for f being Function of X,Y st X c= Z holds f|Z = f;
theorem :: FUNCT_2:34
for f being Function of X,Y st Y <> {} & x in X & f.x in Z holds (Z|`f)
.x = f.x;
theorem :: FUNCT_2:35
for f being Function of X,Y st Y <> {} for y holds (ex x st x in X & x
in P & y = f.x) implies y in f.:P;
theorem :: FUNCT_2:36
for f being Function of X,Y holds f.:P c= Y;
::$CT
theorem :: FUNCT_2:38
for f being Function of X,Y st Y <> {} for x holds x in f"Q iff x in X
& f.x in Q;
theorem :: FUNCT_2:39
for f being PartFunc of X,Y holds f"Q c= X;
theorem :: FUNCT_2:40
for f being Function of X,Y st Y = {} implies X = {} holds f"Y = X;
theorem :: FUNCT_2:41
for f being Function of X,Y holds
(for y being object st y in Y holds f"{y} <> {})
iff rng f = Y;
theorem :: FUNCT_2:42
for f being Function of X,Y st (Y = {} implies X = {}) & P c= X
holds P c= f"(f.:P);
theorem :: FUNCT_2:43
for f being Function of X,Y st Y = {} implies X = {} holds f"(f.:X) = X;
theorem :: FUNCT_2:44
for f being Function of X,Y for g being Function of Y,Z st (Z = {}
implies Y = {}) holds f"Q c= (g*f)"(g.:Q);
theorem :: FUNCT_2:45
for f being Function of {},Y holds f.:P = {};
theorem :: FUNCT_2:46
for f being Function of {},Y holds f"Q = {};
theorem :: FUNCT_2:47
for x being object
for f being Function of {x},Y st Y <> {} holds f.x in Y;
theorem :: FUNCT_2:48
for x being object
for f being Function of {x},Y st Y <> {} holds rng f = {f.x};
theorem :: FUNCT_2:49
for x being object
for f being Function of {x},Y st Y <> {} holds f.:P c= {f.x};
theorem :: FUNCT_2:50
for y being object
for f being Function of X,{y} st x in X holds f.x = y;
theorem :: FUNCT_2:51
for y being object
for f1,f2 being Function of X,{y} holds f1 = f2;
theorem :: FUNCT_2:52
for f being Function of X,X holds dom f = X;
registration
let X,Y be set;
let f be quasi_total PartFunc of X,Y;
let g be quasi_total PartFunc of X,X;
cluster f*g -> quasi_total for PartFunc of X,Y;
end;
registration
let X,Y be set;
let f be quasi_total PartFunc of Y,Y;
let g be quasi_total PartFunc of X,Y;
cluster f*g -> quasi_total for PartFunc of X,Y;
end;
theorem :: FUNCT_2:53
for f,g being Relation of X,X st rng f = X & rng g = X holds rng (g*f) = X;
theorem :: FUNCT_2:54
for f,g being Function of X,X st g*f = f & rng f = X holds g = id X;
theorem :: FUNCT_2:55
for f,g being Function of X,X st f*g = f & f is one-to-one holds g = id X;
theorem :: FUNCT_2:56
for f being Function of X,X holds f is one-to-one iff
for x1,x2 being object
st x1 in X & x2 in X & f.x1 = f.x2 holds x1 = x2;
definition
let X, Y;
let f be X-defined Y-valued Function;
attr f is bijective means
:: FUNCT_2:def 4
f is one-to-one onto;
end;
registration
let X, Y be set;
cluster bijective -> one-to-one onto for PartFunc of X,Y;
cluster one-to-one onto -> bijective for PartFunc of X,Y;
end;
registration
let X;
cluster bijective for Function of X,X;
end;
definition
let X;
mode Permutation of X is bijective Function of X,X;
end;
theorem :: FUNCT_2:57
for f being Function of X, X st f is one-to-one & rng f = X
holds f is Permutation of X;
theorem :: FUNCT_2:58
for f being Function of X,X st f is one-to-one holds for x1,x2 st x1
in X & x2 in X & f.x1 = f.x2 holds x1 = x2;
registration
let X;
let f,g be onto PartFunc of X,X;
cluster f*g -> onto for PartFunc of X,X;
end;
registration
let X;
let f,g be bijective Function of X,X;
cluster g*f -> bijective for Function of X,X;
end;
registration
let X;
cluster reflexive total -> bijective for Function of X,X;
end;
definition
let X;
let f be Permutation of X;
redefine func f" -> Permutation of X;
end;
theorem :: FUNCT_2:59
for f,g being Permutation of X st g*f = g holds f = id X;
theorem :: FUNCT_2:60
for f,g being Permutation of X st g*f = id X holds g = f";
theorem :: FUNCT_2:61
for f being Permutation of X holds (f")*f =id X & f*(f") = id X;
theorem :: FUNCT_2:62
for f being Permutation of X st P c= X holds f.:(f"P) = P & f"(f .:P) = P;
reserve D for non empty set;
registration
let X,D,Z;
let f be Function of X,D;
let g be Function of D,Z;
cluster g*f -> quasi_total for PartFunc of X,Z;
end;
definition
let C be non empty set, D be set;
let f be Function of C,D;
let c be Element of C;
redefine func f.c -> Element of D;
end;
scheme :: FUNCT_2:sch 3
FuncExD{C, D() -> non empty set, P[object,object]}:
ex f being Function of C(),D()
st for x being Element of C() holds P[x,f.x]
provided
for x being Element of C() ex y being Element of D() st P[x,y];
scheme :: FUNCT_2:sch 4
LambdaD{C, D() -> non empty set, F(Element of C()) -> Element of D()}: ex f
being Function of C(),D() st for x being Element of C() holds f.x = F(x);
theorem :: FUNCT_2:63
for f1,f2 being Function of X,Y st for x being Element of X
holds f1.x = f2.x holds f1 = f2;
theorem :: FUNCT_2:64
for P being set for f being Function of X,Y
for y holds y in f.:P implies ex x st x in X & x in P & y = f.x;
theorem :: FUNCT_2:65
for f being Function of X,Y for y st y in f.:P ex c being Element of X
st c in P & y = f.c;
begin :: PARTFUN1
theorem :: FUNCT_2:66
for f being set st f in Funcs(X,Y) holds f is Function of X,Y;
scheme :: FUNCT_2:sch 5
Lambda1C{A, B() -> set, C[object], F, G(object)->object}:
ex f being
Function of A(),B() st
for x being object st x in A() holds (C[ x] implies f.x = F(x)) & (
not C[ x] implies f.x = G(x))
provided
for x being object st x in A()
holds (C[ x] implies F(x) in B()) & (not C[ x]
implies G(x) in B());
theorem :: FUNCT_2:67
for f being PartFunc of X,Y st dom f = X holds f is Function of X,Y;
theorem :: FUNCT_2:68
for f being PartFunc of X,Y st f is total holds f is Function of X,Y;
theorem :: FUNCT_2:69
for f being PartFunc of X,Y st (Y = {} implies X = {}) & f is Function
of X,Y holds f is total;
theorem :: FUNCT_2:70
for f being Function of X,Y st (Y = {} implies X = {}) holds
<:f,X,Y:> is total;
registration
let X; let f be Function of X,X;
cluster <:f,X,X:> -> total;
end;
theorem :: FUNCT_2:71
for f being PartFunc of X,Y st Y = {} implies X = {} ex g being
Function of X,Y st for x being object st x in dom f holds g.x = f.x;
theorem :: FUNCT_2:72
Funcs(X,Y) c= PFuncs(X,Y);
theorem :: FUNCT_2:73
for f,g being Function of X,Y st (Y = {} implies X = {}) & f
tolerates g holds f = g;
theorem :: FUNCT_2:74
for f,g being Function of X,X st f tolerates g holds f = g;
theorem :: FUNCT_2:75
for f being PartFunc of X,Y for g being Function of X,Y st Y =
{} implies X = {} holds f tolerates g iff
for x being object st x in dom f holds f.x = g.x;
theorem :: FUNCT_2:76
for f being PartFunc of X,X for g being Function of X,X holds f
tolerates g iff for x being object st x in dom f holds f.x = g.x;
theorem :: FUNCT_2:77
for f being PartFunc of X,Y st Y = {} implies X = {} ex g being
Function of X,Y st f tolerates g;
theorem :: FUNCT_2:78
for f,g being PartFunc of X,X for h being Function of X,X st f
tolerates h & g tolerates h holds f tolerates g;
theorem :: FUNCT_2:79
for f,g being PartFunc of X,Y st (Y = {} implies X = {}) & f tolerates
g ex h being Function of X,Y st f tolerates h & g tolerates h;
theorem :: FUNCT_2:80
for f being PartFunc of X,Y for g being Function of X,Y st (Y =
{} implies X = {}) & f tolerates g holds g in TotFuncs f;
theorem :: FUNCT_2:81
for f being PartFunc of X,X for g being Function of X,X st f tolerates
g holds g in TotFuncs f;
theorem :: FUNCT_2:82
for f being PartFunc of X,Y for g being set st g in TotFuncs(f)
holds g is Function of X,Y;
theorem :: FUNCT_2:83
for f being PartFunc of X,Y holds TotFuncs f c= Funcs(X,Y);
theorem :: FUNCT_2:84
TotFuncs <:{},X,Y:> = Funcs(X,Y);
theorem :: FUNCT_2:85
for f being Function of X,Y st Y = {} implies X = {} holds
TotFuncs <:f,X,Y:> = {f};
theorem :: FUNCT_2:86
for f being Function of X,X holds TotFuncs <:f,X,X:> = {f};
theorem :: FUNCT_2:87
for f being PartFunc of X,{y} for g being Function of X,{y} holds
TotFuncs f = {g};
theorem :: FUNCT_2:88
for f,g being PartFunc of X,Y st g c= f holds TotFuncs f c= TotFuncs g;
theorem :: FUNCT_2:89
for f,g being PartFunc of X,Y st dom g c= dom f & TotFuncs f c=
TotFuncs g holds g c= f;
theorem :: FUNCT_2:90
for f,g being PartFunc of X,Y st TotFuncs f c= TotFuncs g & (
for y holds Y <> {y}) holds g c= f;
theorem :: FUNCT_2:91
for f,g being PartFunc of X,Y st (for y holds Y <> {y}) & TotFuncs f =
TotFuncs g holds f = g;
:: from WAYBEL_1
registration
let A,B be non empty set;
cluster -> non empty for Function of A,B;
end;
begin :: Addenda
:: from RLVECT_2
scheme :: FUNCT_2:sch 6
LambdaSep1{D, R() -> non empty set, A() -> Element of D(), B() -> Element of
R(), F(object) -> Element of R()}:
ex f being Function of D(),R() st f.A() = B() &
for x being Element of D() st x <> A() holds f.x = F(x);
scheme :: FUNCT_2:sch 7
LambdaSep2{D, R() -> non empty set, A1, A2() -> Element of D(), B1, B2() ->
Element of R(), F(object) -> Element of R()}:
ex f being Function of D(),R() st f.
A1() = B1() & f.A2() = B2() & for x being Element of D() st x <> A1() & x <> A2
() holds f.x = F(x)
provided
A1() <> A2();
:: from ALTCAT_1, PRALG_3
theorem :: FUNCT_2:92
for A,B being set for f being Function st f in Funcs(A,B) holds dom f
= A & rng f c= B;
:: from SUPINF_2
scheme :: FUNCT_2:sch 8
FunctRealEx{X()->non empty set,Y()->set,F(object)->object}:
ex f being Function of
X(),Y() st for x being Element of X() holds f.x = F(x)
provided
for x being Element of X() holds F(x) in Y();
:: from YELLOW_2
scheme :: FUNCT_2:sch 9
KappaMD{X, Y() -> non empty set, F(object) -> object}:
ex f being Function of X(),
Y() st for x being Element of X() holds f.x = F(x)
provided
for x being Element of X() holds F(x) is Element of Y();
definition
let A,B,C be non empty set;
let f be Function of A, [:B,C:];
redefine func pr1 f -> Function of A,B means
:: FUNCT_2:def 5
for x being Element of A holds it.x = (f.x)`1;
redefine func pr2 f -> Function of A,C means
:: FUNCT_2:def 6
for x being Element of A holds it .x = (f.x)`2;
end;
definition
let A1 be set, B1 be non empty set, A2 be set, B2 be non empty set, f1 be
Function of A1,B1, f2 be Function of A2,B2;
redefine pred f1 = f2 means
:: FUNCT_2:def 7
A1 = A2 & for a being Element of A1 holds f1.a = f2.a;
end;
definition
let A,B be set, f1,f2 be Function of A,B;
redefine pred f1 = f2 means
:: FUNCT_2:def 8
for a being Element of A holds f1.a = f2.a;
end;
:: missing, 2006.11.05, A.T.
theorem :: FUNCT_2:93
for N being set, f being Function of N, bool N ex R being Relation of
N st for i being set st i in N holds Im(R,i) = f.i;
:: Moved from BORSUK_1 by AK on 27.12.2006
theorem :: FUNCT_2:94
for A being Subset of X holds (id X)"A = A;
:: Moved from TMAP_1 by AK on 27.12.2006
reserve A,B for non empty set;
theorem :: FUNCT_2:95
for f being Function of A,B, A0 being Subset of A, B0 being Subset of
B holds f.:A0 c= B0 iff A0 c= f"B0;
theorem :: FUNCT_2:96
for f being Function of A,B, A0 being non empty Subset of A, f0 being
Function of A0,B st for c being Element of A st c in A0 holds f.c = f0.c holds
f|A0 = f0;
theorem :: FUNCT_2:97
for f being Function, A0, C being set st C c= A0 holds f.:C = (f|A0).: C;
theorem :: FUNCT_2:98
for f being Function, A0, D being set st f"D c= A0 holds f"D = (f|A0)" D;
scheme :: FUNCT_2:sch 10
MChoice{A()-> non empty set, B()-> non empty set, F(object) -> set}:
ex t being
Function of A(),B() st for a being Element of A() holds t.a in F(a)
provided
for a being Element of A() holds B() meets F(a);
:: Moved from FINSEQ_4 by AK on 2007.10.09
theorem :: FUNCT_2:99
for X, D be non empty set, p be Function of X,D, i be Element
of X holds p/.i = p.i;
registration
let X, D be non empty set, p be Function of X,D, i be Element of X;
identify p/.i with p.i;
end;
:: from JCT_MISC, 2008.06.01, A.T.
theorem :: FUNCT_2:100
for S,X being set, f being Function of S,X, A being Subset of X st X =
{} implies S = {} holds (f"A)` = f"(A`);
:: from CAT_1, 2008.07.01, A.T.
theorem :: FUNCT_2:101
for X,Y,Z being set, D being non empty set, f being Function of X,D st
Y c= X & f.:Y c= Z holds f|Y is Function of Y,Z;
:: from WEIERSTR, 2008.07.07, A.T.
definition
let T,S be non empty set;
let f be Function of T,S;
let G be Subset-Family of S;
func f"G -> Subset-Family of T means
:: FUNCT_2:def 9
for A being Subset of T holds A
in it iff ex B being Subset of S st B in G & A = f"B;
end;
theorem :: FUNCT_2:102
for T,S being non empty set, f being Function of T,S, A,B being
Subset-Family of S st A c= B holds f"A c= f"B;
definition
let T,S be set;
let f be Function of T,S;
let G be Subset-Family of T;
func f.:G -> Subset-Family of S means
:: FUNCT_2:def 10
for A being Subset of S holds
A in it iff ex B being Subset of T st B in G & A = f.:B;
end;
theorem :: FUNCT_2:103
for T,S being set, f being Function of T,S, A,B being Subset-Family of T
st A c= B holds f.:A c= f.:B;
theorem :: FUNCT_2:104
for T,S being non empty set, f being Function of T,S, B being
Subset-Family of S, P being Subset of S st f.:(f"B) is Cover of P holds B is
Cover of P;
theorem :: FUNCT_2:105
for T,S being non empty set, f being Function of T,S, B being
Subset-Family of T, P being Subset of T st B is Cover of P holds f"(f.:B) is
Cover of P;
theorem :: FUNCT_2:106
for T,S being non empty set, f being Function of T,S, Q being
Subset-Family of S holds union(f.:(f"(Q))) c= union Q;
theorem :: FUNCT_2:107
for T,S being non empty set, f being Function of T,S, P being
Subset-Family of T holds union P c= union(f"(f.:P));
:: Generalized RFUNCT_2:def 1,CFCONT_1:def 1,NFCONT_1:def 7,def 8
:: NCFCONT1:def 9,def 10,def 11,def 12,def 13,def 14
:: 2008.08.15, A.T.
definition
let X,Z be set, Y be non empty set;
let f be Function of X,Y;
let p be Z-valued Function;
assume
rng f c= dom p;
func p/*f -> Function of X,Z equals
:: FUNCT_2:def 11
p*f;
end;
reserve Y for non empty set,
f for Function of X,Y,
p for PartFunc of Y,Z,
x for Element of X;
theorem :: FUNCT_2:108
X <> {} & rng f c= dom p implies (p/*f).x = p.(f.x);
theorem :: FUNCT_2:109
X <> {} & rng f c= dom p implies (p/*f).x = p/.(f.x);
reserve g for Function of X,X;
theorem :: FUNCT_2:110
rng f c= dom p implies (p/*f)*g = p/*(f*g);
:: from RAMSEY_1, 2008.09.12,A.T.
theorem :: FUNCT_2:111
for X,Y being non empty set, f being Function of X,Y holds f is
constant iff ex y being Element of Y st rng f = {y};
:: from ISOCAT_2, 2008.09.14, A.T.
theorem :: FUNCT_2:112
for A,B being non empty set, x being Element of A, f being Function of
A,B holds f.x in rng f;
:: missing, 2008.09.14, A.T.
theorem :: FUNCT_2:113
for A,B being set, f being Function of A,B st y in rng f ex x
being Element of A st y = f.x;
:: from RFUNCT_2, 2008.09.14, A.T.
theorem :: FUNCT_2:114
for A,B being non empty set, f being Function of A,B st for x being
Element of A holds f.x in Z holds rng f c= Z;
reserve X,Y for non empty set,
Z,S,T for set,
f for Function of X,Y,
g for PartFunc of Y,Z,
x for Element of X;
theorem :: FUNCT_2:115
g is total implies (g/*f).x = g.(f.x);
theorem :: FUNCT_2:116
g is total implies (g/*f).x = g/.(f.x);
theorem :: FUNCT_2:117
rng f c= dom (g|S) implies (g|S)/*f = g/*f;
theorem :: FUNCT_2:118
rng f c= dom (g|S) & S c= T implies (g|S)/*f = (g|T)/*f;
:: missing, 2009.01.09, A.T.
theorem :: FUNCT_2:119
for H being Function of D, [:A,B:], d being Element of D holds H.d = [
pr1 H.d,pr2 H.d];
:: from YELLOW20, 2009.01.21, A.T.
theorem :: FUNCT_2:120
for A1,A2, B1,B2 being set for f being Function of A1,A2, g being
Function of B1,B2 st f tolerates g holds f /\ g is Function of A1 /\ B1, A2 /\
B2;
:: from FRAENKEL, 2009.05.06, A.K.
registration
let A, B be set;
cluster Funcs(A,B) -> functional;
end;
definition
let A, B be set;
mode FUNCTION_DOMAIN of A,B -> non empty set means
:: FUNCT_2:def 12
for x being Element of it holds x is Function of A,B;
end;
registration
let A, B be set;
cluster -> functional for FUNCTION_DOMAIN of A,B;
end;
theorem :: FUNCT_2:121
for f being Function of P,Q holds { f } is FUNCTION_DOMAIN of P,Q;
theorem :: FUNCT_2:122
Funcs(P,B) is FUNCTION_DOMAIN of P,B;
definition
let A be set, B be non empty set;
redefine func Funcs(A,B) -> FUNCTION_DOMAIN of A,B;
let F be FUNCTION_DOMAIN of A,B;
redefine mode Element of F -> Function of A,B;
end;
registration
let I be set;
cluster id I -> total for I-defined Function;
end;
:: from CAT_3, 2009.09.14
definition
let X,A;
let F be Function of X,A;
let x be set;
assume
x in X;
redefine func F/.x equals
:: FUNCT_2:def 13
F.x;
end;
theorem :: FUNCT_2:123
for X being set, Y being non empty set
for f being Function of X, Y, g being X-valued Function
holds dom(f*g) = dom g;
theorem :: FUNCT_2:124
for X being non empty set, f being Function of X,X st for x being
Element of X holds f.x = x holds f = id X;
:: Moved from GROUP_9, AK
definition
let O,E be set;
mode Action of O,E is Function of O, Funcs(E,E);
end;
theorem :: FUNCT_2:125
for x being set, A being set
for f,g being Function of {x}, A st f.x = g.x
holds f = g;
theorem :: FUNCT_2:126
for A being set holds id A in Funcs(A,A);
theorem :: FUNCT_2:127
Funcs({},{}) = { id {} };
theorem :: FUNCT_2:128
for A,B,C being set, f,g being Function st f in Funcs(A,B) & g in
Funcs(B,C) holds g*f in Funcs(A,C);
theorem :: FUNCT_2:129
for A,B,C being set st Funcs(A,B) <> {} & Funcs(B,C) <> {} holds
Funcs(A,C) <> {};
scheme :: FUNCT_2:sch 11
Lambda1{X, Y() -> set, F(object)->object}:
ex f being Function of X(),Y() st
for x being set st x in X() holds f.x = F(x)
provided
for x being set st x in X() holds F(x) in Y();
scheme :: FUNCT_2:sch 12
Lambda1{X, Y() -> set, F(object)-> object}:
ex f being Function of X(),Y() st
for x being set st x in X() holds f.x = F(x)
provided
for x being set st x in X() holds F(x) in Y();
theorem :: FUNCT_2:130
for A being set holds {} is Function of A,{};