:: Functions and Their Basic Properties
:: by Czes{\l}aw Byli\'nski
::
:: Received March 3, 1989
:: Copyright (c) 1990-2011 Association of Mizar Users


begin

definition
let X be set ;
attr X is Function-like means :Def1: :: FUNCT_1:def 1
 for x, y1, y2 being set st [x,y1] in X & [x,y2] in X holds
y1 = y2;
end;

:: deftheorem Def1 defines Function-like FUNCT_1:def 1 :
for X being set holds
( X is Function-like iff for x, y1, y2 being set st [x,y1] in X & [x,y2] in X holds
y1 = y2 );

registration
cluster empty -> Function-like set ;
coherence
for b1 being set st b1 is empty holds
b1 is Function-like
proof end;
end;

registration
cluster Relation-like Function-like set ;
existence
ex b1 being Relation st b1 is Function-like
proof end;
end;

definition
mode Function is Function-like Relation;
end;

registration
let a, b be set ;
cluster {[a,b]} -> Function-like ;
coherence
{[a,b]} is Function-like
proof end;
end;

theorem :: FUNCT_1:1
canceled;

theorem :: FUNCT_1:2
canceled;

scheme :: FUNCT_1:sch 1
GraphFunc{ F1() -> set , P1[ set , set ] } :
ex f being Function st
for x, y being set holds
( [x,y] in f iff ( x in F1() & P1[x,y] ) )
provided
A1: for x, y1, y2 being set st P1[x,y1] & P1[x,y2] holds
y1 = y2
proof end;

definition
let f be Function;
let x be set ;
canceled;
canceled;
func f . x -> set means :Def4: :: FUNCT_1:def 4
[x,it] in f if x in dom f
otherwise it = {} ;
existence
( ( x in dom f implies ex b1 being set st [x,b1] in f ) & ( not x in dom f implies ex b1 being set st b1 = {} ) ) by RELAT_1:def 4;
uniqueness
for b1, b2 being set holds
( ( x in dom f & [x,b1] in f & [x,b2] in f implies b1 = b2 ) & ( not x in dom f & b1 = {} & b2 = {} implies b1 = b2 ) ) by Def1;
consistency
for b1 being set holds verum ;
end;

:: deftheorem FUNCT_1:def 2 :
canceled;

:: deftheorem FUNCT_1:def 3 :
canceled;

:: deftheorem Def4 defines . FUNCT_1:def 4 :
for f being Function
for x, b3 being set holds
( ( x in dom f implies ( b3 = f . x iff [x,b3] in f ) ) & ( not x in dom f implies ( b3 = f . x iff b3 = {} ) ) );

theorem :: FUNCT_1:3
canceled;

theorem :: FUNCT_1:4
canceled;

theorem :: FUNCT_1:5
canceled;

theorem :: FUNCT_1:6
canceled;

theorem :: FUNCT_1:7
canceled;

theorem Th8: :: FUNCT_1:8
for x, y being set
for f being Function holds
( [x,y] in f iff ( x in dom f & y = f . x ) )
proof end;

theorem Th9: :: FUNCT_1:9
for f, g being Function st dom f = dom g & ( for x being set st x in dom f holds
f . x = g . x ) holds
f = g
proof end;

definition
let f be Function;
redefine func proj2 f means :Def5: :: FUNCT_1:def 5
 for y being set holds
( y in it iff ex x being set st
( x in dom f & y = f . x ) );
compatibility
for b1 being set holds
( b1 = rng f iff for y being set holds
( y in b1 iff ex x being set st
( x in dom f & y = f . x ) ) )
proof end;
end;

:: deftheorem Def5 defines rng FUNCT_1:def 5 :
for f being Function
for b2 being set holds
( b2 = rng f iff for y being set holds
( y in b2 iff ex x being set st
( x in dom f & y = f . x ) ) );

theorem :: FUNCT_1:10
canceled;

theorem :: FUNCT_1:11
canceled;

theorem Th12: :: FUNCT_1:12
for x being set
for f being Function st x in dom f holds
f . x in rng f by Def5;

theorem :: FUNCT_1:13
canceled;

theorem Th14: :: FUNCT_1:14
for x being set
for f being Function st dom f = {x} holds
rng f = {(f . x)}
proof end;

scheme :: FUNCT_1:sch 2
FuncEx{ F1() -> set , P1[ set , set ] } :
ex f being Function st
( dom f = F1() & ( for x being set st x in F1() holds
P1[x,f . x] ) )
provided
A1: for x, y1, y2 being set st x in F1() & P1[x,y1] & P1[x,y2] holds
y1 = y2 and
A2: for x being set st x in F1() holds
ex y being set st P1[x,y]
proof end;

scheme :: FUNCT_1:sch 3
Lambda{ F1() -> set , F2( set ) -> set } :
ex f being Function st
( dom f = F1() & ( for x being set st x in F1() holds
f . x = F2(x) ) )
proof end;

theorem Th15: :: FUNCT_1:15
for X being set st X <> {} holds
for y being set ex f being Function st
( dom f = X & rng f = {y} )
proof end;

theorem :: FUNCT_1:16
for X being set st ( for f, g being Function st dom f = X & dom g = X holds
f = g ) holds
X = {}
proof end;

theorem :: FUNCT_1:17
for y being set
for f, g being Function st dom f = dom g & rng f = {y} & rng g = {y} holds
f = g
proof end;

theorem :: FUNCT_1:18
for Y, X being set st ( Y <> {} or X = {} ) holds
ex f being Function st
( X = dom f & rng f c= Y )
proof end;

theorem :: FUNCT_1:19
for Y being set
for f being Function st ( for y being set st y in Y holds
ex x being set st
( x in dom f & y = f . x ) ) holds
Y c= rng f
proof end;

notation
let f, g be Function;
synonym g * f for f * g;
end;

registration
let f, g be Function;
cluster g * f -> Function-like ;
coherence
g * f is Function-like
proof end;
end;

theorem :: FUNCT_1:20
for f, g, h being Function st ( for x being set holds
( x in dom h iff ( x in dom f & f . x in dom g ) ) ) & ( for x being set st x in dom h holds
h . x = g . (f . x) ) holds
h = g * f
proof end;

theorem Th21: :: FUNCT_1:21
for x being set
for g, f being Function holds
( x in dom (g * f) iff ( x in dom f & f . x in dom g ) )
proof end;

theorem Th22: :: FUNCT_1:22
for x being set
for g, f being Function st x in dom (g * f) holds
(g * f) . x = g . (f . x)
proof end;

theorem Th23: :: FUNCT_1:23
for x being set
for f, g being Function st x in dom f holds
(g * f) . x = g . (f . x)
proof end;

theorem :: FUNCT_1:24
canceled;

theorem :: FUNCT_1:25
for z being set
for g, f being Function st z in rng (g * f) holds
z in rng g
proof end;

theorem :: FUNCT_1:26
canceled;

theorem Th27: :: FUNCT_1:27
for g, f being Function st dom (g * f) = dom f holds
rng f c= dom g
proof end;

theorem :: FUNCT_1:28
canceled;

theorem :: FUNCT_1:29
canceled;

theorem :: FUNCT_1:30
canceled;

theorem :: FUNCT_1:31
canceled;

theorem :: FUNCT_1:32
canceled;

theorem :: FUNCT_1:33
for Y being set
for f being Function st rng f c= Y & ( for g, h being Function st dom g = Y & dom h = Y & g * f = h * f holds
g = h ) holds
Y = rng f
proof end;

registration
let X be set ;
cluster id X -> Function-like ;
coherence
id X is Function-like
proof end;
end;

theorem Th34: :: FUNCT_1:34
for X being set
for f being Function holds
( f = id X iff ( dom f = X & ( for x being set st x in X holds
f . x = x ) ) )
proof end;

theorem :: FUNCT_1:35
for x, X being set st x in X holds
(id X) . x = x by Th34;

theorem :: FUNCT_1:36
canceled;

theorem Th37: :: FUNCT_1:37
for X being set
for f being Function holds dom (f * (id X)) = (dom f) /\ X
proof end;

theorem :: FUNCT_1:38
for x, X being set
for f being Function st x in (dom f) /\ X holds
f . x = (f * (id X)) . x
proof end;

theorem :: FUNCT_1:39
canceled;

theorem :: FUNCT_1:40
for x, Y being set
for f being Function holds
( x in dom ((id Y) * f) iff ( x in dom f & f . x in Y ) )
proof end;

theorem :: FUNCT_1:41
canceled;

theorem :: FUNCT_1:42
canceled;

theorem :: FUNCT_1:43
for X, Y being set holds (id X) * (id Y) = id (X /\ Y)
proof end;

theorem Th44: :: FUNCT_1:44
for f, g being Function st rng f = dom g & g * f = f holds
g = id (dom g)
proof end;

definition
let f be Function;
canceled;
canceled;
attr f is one-to-one means :Def8: :: FUNCT_1:def 8
 for x1, x2 being set st x1 in dom f & x2 in dom f & f . x1 = f . x2 holds
x1 = x2;
end;

:: deftheorem FUNCT_1:def 6 :
canceled;

:: deftheorem FUNCT_1:def 7 :
canceled;

:: deftheorem Def8 defines one-to-one FUNCT_1:def 8 :
for f being Function holds
( f is one-to-one iff for x1, x2 being set st x1 in dom f & x2 in dom f & f . x1 = f . x2 holds
x1 = x2 );

theorem :: FUNCT_1:45
canceled;

theorem Th46: :: FUNCT_1:46
for f, g being Function st f is one-to-one & g is one-to-one holds
g * f is one-to-one
proof end;

theorem Th47: :: FUNCT_1:47
for g, f being Function st g * f is one-to-one & rng f c= dom g holds
f is one-to-one
proof end;

theorem :: FUNCT_1:48
for g, f being Function st g * f is one-to-one & rng f = dom g holds
( f is one-to-one & g is one-to-one )
proof end;

theorem :: FUNCT_1:49
for f being Function holds
( f is one-to-one iff for g, h being Function st rng g c= dom f & rng h c= dom f & dom g = dom h & f * g = f * h holds
g = h )
proof end;

theorem :: FUNCT_1:50
for X being set
for f, g being Function st dom f = X & dom g = X & rng g c= X & f is one-to-one & f * g = f holds
g = id X
proof end;

theorem :: FUNCT_1:51
for g, f being Function st rng (g * f) = rng g & g is one-to-one holds
dom g c= rng f
proof end;

registration
let X be set ;
cluster id X -> one-to-one ;
coherence
id X is one-to-one
proof end;
end;

theorem :: FUNCT_1:52
for X being set holds id X is one-to-one ;

theorem :: FUNCT_1:53
for f being Function st ex g being Function st g * f = id (dom f) holds
f is one-to-one
proof end;

registration
cluster empty Relation-like Function-like -> one-to-one set ;
coherence
for b1 being Function st b1 is empty holds
b1 is one-to-one
proof end;
end;

registration
cluster Relation-like Function-like one-to-one set ;
existence
ex b1 being Function st b1 is one-to-one
proof end;
end;

registration
let f be one-to-one Function;
cluster f ~ -> Function-like ;
coherence
f ~ is Function-like
proof end;
end;

definition
let f be Function;
assume A1: f is one-to-one ;
func f " -> Function equals :Def9: :: FUNCT_1:def 9
f ~ ;
coherence
f ~ is Function by A1;
end;

:: deftheorem Def9 defines " FUNCT_1:def 9 :
for f being Function st f is one-to-one holds
f " = f ~ ;

theorem Th54: :: FUNCT_1:54
for f being Function st f is one-to-one holds
for g being Function holds
( g = f " iff ( dom g = rng f & ( for y, x being set holds
( y in rng f & x = g . y iff ( x in dom f & y = f . x ) ) ) ) )
proof end;

theorem Th55: :: FUNCT_1:55
for f being Function st f is one-to-one holds
( rng f = dom (f ") & dom f = rng (f ") )
proof end;

theorem Th56: :: FUNCT_1:56
for x being set
for f being Function st f is one-to-one & x in dom f holds
( x = (f ") . (f . x) & x = ((f ") * f) . x )
proof end;

theorem Th57: :: FUNCT_1:57
for y being set
for f being Function st f is one-to-one & y in rng f holds
( y = f . ((f ") . y) & y = (f * (f ")) . y )
proof end;

theorem Th58: :: FUNCT_1:58
for f being Function st f is one-to-one holds
( dom ((f ") * f) = dom f & rng ((f ") * f) = dom f )
proof end;

theorem Th59: :: FUNCT_1:59
for f being Function st f is one-to-one holds
( dom (f * (f ")) = rng f & rng (f * (f ")) = rng f )
proof end;

theorem :: FUNCT_1:60
for f, g being Function st f is one-to-one & dom f = rng g & rng f = dom g & ( for x, y being set st x in dom f & y in dom g holds
( f . x = y iff g . y = x ) ) holds
g = f "
proof end;

theorem Th61: :: FUNCT_1:61
for f being Function st f is one-to-one holds
( (f ") * f = id (dom f) & f * (f ") = id (rng f) )
proof end;

theorem Th62: :: FUNCT_1:62
for f being Function st f is one-to-one holds
f " is one-to-one
proof end;

registration
let f be one-to-one Function;
cluster f " -> one-to-one ;
coherence
f " is one-to-one by Th62;
let g be one-to-one Function;
cluster g * f -> one-to-one ;
coherence
g * f is one-to-one by Th46;
end;

Lm1: for X being set
for g2, f, g1 being Function st rng g2 = X & f * g2 = id (dom g1) & g1 * f = id X holds
g1 = g2
proof end;

theorem Th63: :: FUNCT_1:63
for f, g being Function st f is one-to-one & rng f = dom g & g * f = id (dom f) holds
g = f "
proof end;

theorem :: FUNCT_1:64
for f, g being Function st f is one-to-one & rng g = dom f & f * g = id (rng f) holds
g = f "
proof end;

theorem :: FUNCT_1:65
for f being Function st f is one-to-one holds
(f ") " = f
proof end;

theorem :: FUNCT_1:66
for f, g being Function st f is one-to-one & g is one-to-one holds
(g * f) " = (f ") * (g ")
proof end;

theorem :: FUNCT_1:67
for X being set holds (id X) " = id X
proof end;

registration
let f be Function;
let X be set ;
cluster f | X -> Function-like ;
coherence
f | X is Function-like
proof end;
end;

theorem :: FUNCT_1:68
for X being set
for g, f being Function st dom g = (dom f) /\ X & ( for x being set st x in dom g holds
g . x = f . x ) holds
g = f | X
proof end;

theorem :: FUNCT_1:69
canceled;

theorem Th70: :: FUNCT_1:70
for x, X being set
for f being Function st x in dom (f | X) holds
(f | X) . x = f . x
proof end;

theorem Th71: :: FUNCT_1:71
for x, X being set
for f being Function st x in (dom f) /\ X holds
(f | X) . x = f . x
proof end;

theorem Th72: :: FUNCT_1:72
for x, X being set
for f being Function st x in X holds
(f | X) . x = f . x
proof end;

theorem :: FUNCT_1:73
for x, X being set
for f being Function st x in dom f & x in X holds
f . x in rng (f | X)
proof end;

theorem :: FUNCT_1:74
canceled;

theorem :: FUNCT_1:75
canceled;

theorem :: FUNCT_1:76
canceled;

theorem :: FUNCT_1:77
canceled;

theorem :: FUNCT_1:78
canceled;

theorem :: FUNCT_1:79
canceled;

theorem :: FUNCT_1:80
canceled;

theorem :: FUNCT_1:81
canceled;

theorem :: FUNCT_1:82
for X, Y being set
for f being Function st X c= Y holds
( (f | X) | Y = f | X & (f | Y) | X = f | X ) by RELAT_1:102, RELAT_1:103;

theorem :: FUNCT_1:83
canceled;

theorem :: FUNCT_1:84
for X being set
for f being Function st f is one-to-one holds
f | X is one-to-one
proof end;

registration
let Y be set ;
let f be Function;
cluster Y | f -> Function-like ;
coherence
Y | f is Function-like
proof end;
end;

theorem Th85: :: FUNCT_1:85
for Y being set
for g, f being Function holds
( g = Y | f iff ( ( for x being set holds
( x in dom g iff ( x in dom f & f . x in Y ) ) ) & ( for x being set st x in dom g holds
g . x = f . x ) ) )
proof end;

theorem :: FUNCT_1:86
for x, Y being set
for f being Function holds
( x in dom (Y | f) iff ( x in dom f & f . x in Y ) ) by Th85;

theorem :: FUNCT_1:87
for x, Y being set
for f being Function st x in dom (Y | f) holds
(Y | f) . x = f . x by Th85;

theorem :: FUNCT_1:88
canceled;

theorem :: FUNCT_1:89
for Y being set
for f being Function holds dom (Y | f) c= dom f
proof end;

theorem :: FUNCT_1:90
canceled;

theorem :: FUNCT_1:91
canceled;

theorem :: FUNCT_1:92
canceled;

theorem :: FUNCT_1:93
canceled;

theorem :: FUNCT_1:94
canceled;

theorem :: FUNCT_1:95
canceled;

theorem :: FUNCT_1:96
canceled;

theorem :: FUNCT_1:97
for X, Y being set
for f being Function st X c= Y holds
( Y | (X | f) = X | f & X | (Y | f) = X | f ) by RELAT_1:129, RELAT_1:130;

theorem :: FUNCT_1:98
canceled;

theorem :: FUNCT_1:99
for Y being set
for f being Function st f is one-to-one holds
Y | f is one-to-one
proof end;

definition
let f be Function;
let X be set ;
canceled;
canceled;
redefine func f .: X means :Def12: :: FUNCT_1:def 12
 for y being set holds
( y in it iff ex x being set st
( x in dom f & x in X & y = f . x ) );
compatibility
for b1 being set holds
( b1 = f .: X iff for y being set holds
( y in b1 iff ex x being set st
( x in dom f & x in X & y = f . x ) ) )
proof end;
end;

:: deftheorem FUNCT_1:def 10 :
canceled;

:: deftheorem FUNCT_1:def 11 :
canceled;

:: deftheorem Def12 defines .: FUNCT_1:def 12 :
for f being Function
for X being set
for b3 being set holds
( b3 = f .: X iff for y being set holds
( y in b3 iff ex x being set st
( x in dom f & x in X & y = f . x ) ) );

theorem :: FUNCT_1:100
canceled;

theorem :: FUNCT_1:101
canceled;

theorem :: FUNCT_1:102
canceled;

theorem :: FUNCT_1:103
canceled;

theorem :: FUNCT_1:104
canceled;

theorem :: FUNCT_1:105
canceled;

theorem :: FUNCT_1:106
canceled;

theorem :: FUNCT_1:107
canceled;

theorem :: FUNCT_1:108
canceled;

theorem :: FUNCT_1:109
canceled;

theorem :: FUNCT_1:110
canceled;

theorem :: FUNCT_1:111
canceled;

theorem :: FUNCT_1:112
canceled;

theorem :: FUNCT_1:113
canceled;

theorem :: FUNCT_1:114
canceled;

theorem :: FUNCT_1:115
canceled;

theorem :: FUNCT_1:116
canceled;

theorem Th117: :: FUNCT_1:117
for x being set
for f being Function st x in dom f holds
Im (f,x) = {(f . x)}
proof end;

theorem :: FUNCT_1:118
for x1, x2 being set
for f being Function st x1 in dom f & x2 in dom f holds
f .: {x1,x2} = {(f . x1),(f . x2)}
proof end;

theorem :: FUNCT_1:119
canceled;

theorem :: FUNCT_1:120
for Y, X being set
for f being Function holds (Y | f) .: X c= f .: X
proof end;

theorem Th121: :: FUNCT_1:121
for X1, X2 being set
for f being Function st f is one-to-one holds
f .: (X1 /\ X2) = (f .: X1) /\ (f .: X2)
proof end;

theorem :: FUNCT_1:122
for f being Function st ( for X1, X2 being set holds f .: (X1 /\ X2) = (f .: X1) /\ (f .: X2) ) holds
f is one-to-one
proof end;

theorem :: FUNCT_1:123
for X1, X2 being set
for f being Function st f is one-to-one holds
f .: (X1 \ X2) = (f .: X1) \ (f .: X2)
proof end;

theorem :: FUNCT_1:124
for f being Function st ( for X1, X2 being set holds f .: (X1 \ X2) = (f .: X1) \ (f .: X2) ) holds
f is one-to-one
proof end;

theorem :: FUNCT_1:125
for X, Y being set
for f being Function st X misses Y & f is one-to-one holds
f .: X misses f .: Y
proof end;

theorem :: FUNCT_1:126
for Y, X being set
for f being Function holds (Y | f) .: X = Y /\ (f .: X)
proof end;

definition
let f be Function;
let Y be set ;
redefine func f " Y means :Def13: :: FUNCT_1:def 13
 for x being set holds
( x in it iff ( x in dom f & f . x in Y ) );
compatibility
for b1 being set holds
( b1 = f " Y iff for x being set holds
( x in b1 iff ( x in dom f & f . x in Y ) ) )
proof end;
end;

:: deftheorem Def13 defines " FUNCT_1:def 13 :
for f being Function
for Y being set
for b3 being set holds
( b3 = f " Y iff for x being set holds
( x in b3 iff ( x in dom f & f . x in Y ) ) );

theorem :: FUNCT_1:127
canceled;

theorem :: FUNCT_1:128
canceled;

theorem :: FUNCT_1:129
canceled;

theorem :: FUNCT_1:130
canceled;

theorem :: FUNCT_1:131
canceled;

theorem :: FUNCT_1:132
canceled;

theorem :: FUNCT_1:133
canceled;

theorem :: FUNCT_1:134
canceled;

theorem :: FUNCT_1:135
canceled;

theorem :: FUNCT_1:136
canceled;

theorem Th137: :: FUNCT_1:137
for Y1, Y2 being set
for f being Function holds f " (Y1 /\ Y2) = (f " Y1) /\ (f " Y2)
proof end;

theorem :: FUNCT_1:138
for Y1, Y2 being set
for f being Function holds f " (Y1 \ Y2) = (f " Y1) \ (f " Y2)
proof end;

theorem :: FUNCT_1:139
for X, Y being set
for R being Relation holds (R | X) " Y = X /\ (R " Y)
proof end;

theorem :: FUNCT_1:140
canceled;

theorem :: FUNCT_1:141
for f being Function
for A, B being set st A misses B holds
f " A misses f " B
proof end;

theorem Th142: :: FUNCT_1:142
for y being set
for R being Relation holds
( y in rng R iff R " {y} <> {} )
proof end;

theorem :: FUNCT_1:143
for Y being set
for R being Relation st ( for y being set st y in Y holds
R " {y} <> {} ) holds
Y c= rng R
proof end;

theorem Th144: :: FUNCT_1:144
for f being Function holds
( ( for y being set st y in rng f holds
ex x being set st f " {y} = {x} ) iff f is one-to-one )
proof end;

theorem Th145: :: FUNCT_1:145
for Y being set
for f being Function holds f .: (f " Y) c= Y
proof end;

theorem Th146: :: FUNCT_1:146
for X being set
for R being Relation st X c= dom R holds
X c= R " (R .: X)
proof end;

theorem :: FUNCT_1:147
for Y being set
for f being Function st Y c= rng f holds
f .: (f " Y) = Y
proof end;

theorem :: FUNCT_1:148
for Y being set
for f being Function holds f .: (f " Y) = Y /\ (f .: (dom f))
proof end;

theorem Th149: :: FUNCT_1:149
for X, Y being set
for f being Function holds f .: (X /\ (f " Y)) c= (f .: X) /\ Y
proof end;

theorem :: FUNCT_1:150
for X, Y being set
for f being Function holds f .: (X /\ (f " Y)) = (f .: X) /\ Y
proof end;

theorem :: FUNCT_1:151
for X, Y being set
for R being Relation holds X /\ (R " Y) c= R " ((R .: X) /\ Y)
proof end;

theorem Th152: :: FUNCT_1:152
for X being set
for f being Function st f is one-to-one holds
f " (f .: X) c= X
proof end;

theorem :: FUNCT_1:153
for f being Function st ( for X being set holds f " (f .: X) c= X ) holds
f is one-to-one
proof end;

theorem :: FUNCT_1:154
for X being set
for f being Function st f is one-to-one holds
f .: X = (f ") " X
proof end;

theorem :: FUNCT_1:155
for Y being set
for f being Function st f is one-to-one holds
f " Y = (f ") .: Y
proof end;

theorem :: FUNCT_1:156
for Y being set
for f, g, h being Function st Y = rng f & dom g = Y & dom h = Y & g * f = h * f holds
g = h
proof end;

theorem :: FUNCT_1:157
for X1, X2 being set
for f being Function st f .: X1 c= f .: X2 & X1 c= dom f & f is one-to-one holds
X1 c= X2
proof end;

theorem Th158: :: FUNCT_1:158
for Y1, Y2 being set
for f being Function st f " Y1 c= f " Y2 & Y1 c= rng f holds
Y1 c= Y2
proof end;

theorem :: FUNCT_1:159
for f being Function holds
( f is one-to-one iff for y being set ex x being set st f " {y} c= {x} )
proof end;

theorem :: FUNCT_1:160
for X being set
for R, S being Relation st rng R c= dom S holds
R " X c= (R * S) " (S .: X)
proof end;

theorem :: FUNCT_1:161
for X, Y being set
for f being Function st f " X = f " Y & X c= rng f & Y c= rng f holds
X = Y
proof end;

begin

theorem :: FUNCT_1:162
for X being set
for A being Subset of X holds (id X) .: A = A
proof end;

definition
let f be Function;
redefine attr f is empty-yielding means :: FUNCT_1:def 14
 for x being set st x in dom f holds
f . x is empty ;
compatibility
( f is empty-yielding iff for x being set st x in dom f holds
f . x is empty )
proof end;
end;

:: deftheorem defines empty-yielding FUNCT_1:def 14 :
for f being Function holds
( f is empty-yielding iff for x being set st x in dom f holds
f . x is empty );

definition
let F be Function;
redefine attr F is non-empty means :Def15: :: FUNCT_1:def 15
 for n being set st n in dom F holds
not F . n is empty ;
compatibility
( F is non-empty iff for n being set st n in dom F holds
not F . n is empty )
proof end;
end;

:: deftheorem Def15 defines non-empty FUNCT_1:def 15 :
for F being Function holds
( F is non-empty iff for n being set st n in dom F holds
not F . n is empty );

registration
cluster Relation-like non-empty Function-like set ;
existence
ex b1 being Function st b1 is non-empty
proof end;
end;

scheme :: FUNCT_1:sch 4
LambdaB{ F1() -> non empty set , F2( set ) -> set } :
ex f being Function st
( dom f = F1() & ( for d being Element of F1() holds f . d = F2(d) ) )
proof end;

registration
let f be non-empty Function;
cluster rng f -> with_non-empty_elements ;
coherence
rng f is with_non-empty_elements
proof end;
end;

definition
let f be Function;
attr f is constant means :Def16: :: FUNCT_1:def 16
 for x, y being set st x in dom f & y in dom f holds
f . x = f . y;
end;

:: deftheorem Def16 defines constant FUNCT_1:def 16 :
for f being Function holds
( f is constant iff for x, y being set st x in dom f & y in dom f holds
f . x = f . y );

theorem :: FUNCT_1:163
for A, B being set
for f being Function st A c= dom f & f .: A c= B holds
A c= f " B
proof end;

theorem :: FUNCT_1:164
for X being set
for f being Function st X c= dom f & f is one-to-one holds
f " (f .: X) = X
proof end;

definition
let f, g be Function;
redefine pred f = g means :: FUNCT_1:def 17
( dom f = dom g & ( for x being set st x in dom f holds
f . x = g . x ) );
compatibility
( f = g iff ( dom f = dom g & ( for x being set st x in dom f holds
f . x = g . x ) ) ) by Th9;
end;

:: deftheorem defines = FUNCT_1:def 17 :
for f, g being Function holds
( f = g iff ( dom f = dom g & ( for x being set st x in dom f holds
f . x = g . x ) ) );

registration
cluster non empty Relation-like non-empty Function-like set ;
existence
ex b1 being Function st
( b1 is non-empty & not b1 is empty )
proof end;
end;

registration
let a be non empty non-empty Function;
let i be Element of dom a;
cluster a . i -> non empty ;
coherence
not a . i is empty
proof end;
end;

registration
let f be Function;
cluster -> Function-like Element of bool f;
coherence
for b1 being Subset of f holds b1 is Function-like
proof end;
end;

theorem :: FUNCT_1:165
for f, g being Function
for D being set st D c= dom f & D c= dom g holds
( f | D = g | D iff for x being set st x in D holds
f . x = g . x )
proof end;

theorem :: FUNCT_1:166
for f, g being Function
for X being set st dom f = dom g & ( for x being set st x in X holds
f . x = g . x ) holds
f | X = g | X
proof end;

theorem Th167: :: FUNCT_1:167
for X being set
for f being Function holds rng (f | {X}) c= {(f . X)}
proof end;

theorem :: FUNCT_1:168
for X being set
for f being Function st X in dom f holds
rng (f | {X}) = {(f . X)}
proof end;

registration
cluster empty Relation-like Function-like -> constant set ;
coherence
for b1 being Function st b1 is empty holds
b1 is constant
proof end;
end;

registration
let X be trivial set ;
cluster -> trivial Element of bool X;
coherence
for b1 being Subset of X holds b1 is trivial
proof end;
end;

registration
let f be constant Function;
cluster rng f -> trivial ;
coherence
rng f is trivial
proof end;
end;

registration
cluster Relation-like Function-like non constant set ;
existence
not for b1 being Function holds b1 is constant
proof end;
end;

registration
let f be non constant Function;
cluster rng f -> non trivial ;
coherence
not rng f is trivial
proof end;
end;

registration
cluster Relation-like Function-like non constant -> non trivial set ;
coherence
for b1 being Function st not b1 is constant holds
not b1 is trivial
proof end;
end;

registration
cluster trivial Relation-like Function-like -> constant set ;
coherence
for b1 being Function st b1 is trivial holds
b1 is constant ;
end;

theorem :: FUNCT_1:169
for F, G being Function
for X being set holds (G | (F .: X)) * (F | X) = (G * F) | X
proof end;

theorem :: FUNCT_1:170
for F, G being Function
for X, X1 being set holds (G | X1) * (F | X) = (G * F) | (X /\ (F " X1))
proof end;

theorem :: FUNCT_1:171
for F, G being Function
for X being set holds
( X c= dom (G * F) iff ( X c= dom F & F .: X c= dom G ) )
proof end;

definition
let f be Function;
assume A1: ( not f is empty & f is constant ) ;
func the_value_of f -> set means :: FUNCT_1:def 18
 ex x being set st
( x in dom f & it = f . x );
existence
ex b1 being set ex x being set st
( x in dom f & b1 = f . x )
proof end;
uniqueness
for b1, b2 being set st ex x being set st
( x in dom f & b1 = f . x ) & ex x being set st
( x in dom f & b2 = f . x ) holds
b1 = b2 by A1, Def16;
end;

:: deftheorem defines the_value_of FUNCT_1:def 18 :
for f being Function st not f is empty & f is constant holds
for b2 being set holds
( b2 = the_value_of f iff ex x being set st
( x in dom f & b2 = f . x ) );

registration
let X, Y be set ;
cluster Relation-like X -defined Y -valued Function-like set ;
existence
ex b1 being Function st
( b1 is X -defined & b1 is Y -valued )
proof end;
end;

theorem :: FUNCT_1:172
for X being set
for f being b1 -valued Function
for x being set st x in dom f holds
f . x in X
proof end;

definition
let IT be set ;
attr IT is functional means :Def19: :: FUNCT_1:def 19
 for x being set st x in IT holds
x is Function;
end;

:: deftheorem Def19 defines functional FUNCT_1:def 19 :
for IT being set holds
( IT is functional iff for x being set st x in IT holds
x is Function );

registration
cluster empty -> functional set ;
coherence
for b1 being set st b1 is empty holds
b1 is functional
proof end;
let f be Function;
cluster {f} -> functional ;
coherence
{f} is functional
proof end;
let g be Function;
cluster {f,g} -> functional ;
coherence
{f,g} is functional
proof end;
end;

registration
cluster non empty functional set ;
existence
ex b1 being set st
( not b1 is empty & b1 is functional )
proof end;
end;

registration
let P be functional set ;
cluster -> Relation-like Function-like Element of P;
coherence
for b1 being Element of P holds
( b1 is Function-like & b1 is Relation-like )
proof end;
end;

registration
let A be functional set ;
cluster -> functional Element of bool A;
coherence
for b1 being Subset of A holds b1 is functional
proof end;
end;

definition
let g, f be Function;
attr f is g -compatible means :Def20: :: FUNCT_1:def 20
 for x being set st x in dom f holds
f . x in g . x;
end;

:: deftheorem Def20 defines -compatible FUNCT_1:def 20 :
for g, f being Function holds
( f is g -compatible iff for x being set st x in dom f holds
f . x in g . x );

theorem :: FUNCT_1:173
for f, g being Function st f is g -compatible & dom f = dom g holds
g is non-empty
proof end;

theorem Th174: :: FUNCT_1:174
for f being Function holds {} is f -compatible
proof end;

registration
let I be set ;
let f be Function;
cluster empty Relation-like I -defined Function-like f -compatible set ;
existence
ex b1 being Function st
( b1 is empty & b1 is I -defined & b1 is f -compatible )
proof end;
end;

registration
let X be set ;
let f be Function;
let g be f -compatible Function;
cluster g | X -> f -compatible ;
coherence
g | X is f -compatible
proof end;
end;

registration
let I be set ;
cluster Relation-like non-empty I -defined Function-like set ;
existence
ex b1 being Function st
( b1 is non-empty & b1 is I -defined )
proof end;
end;

theorem Th175: :: FUNCT_1:175
for f being Function
for g being b1 -compatible Function holds dom g c= dom f
proof end;

registration
let X be set ;
let f be X -defined Function;
cluster Relation-like Function-like f -compatible -> X -defined set ;
coherence
for b1 being Function st b1 is f -compatible holds
b1 is X -defined
proof end;
end;

theorem :: FUNCT_1:176
for X, x being set
for f being b1 -valued Function st x in dom f holds
f . x is Element of X
proof end;

theorem :: FUNCT_1:177
for f being Function
for A being set st f is one-to-one & A c= dom f holds
(f ") .: (f .: A) = A
proof end;

registration
let A be functional set ;
let x be set ;
let F be A -valued Function;
cluster F . x -> Relation-like Function-like ;
coherence
( F . x is Function-like & F . x is Relation-like )
proof end;
end;