:: GRFUNC_1 semantic presentation

theorem Th1: :: GRFUNC_1:1
canceled;

theorem Th2: :: GRFUNC_1:2
canceled;

theorem Th3: :: GRFUNC_1:3
canceled;

theorem Th4: :: GRFUNC_1:4
canceled;

theorem Th5: :: GRFUNC_1:5
canceled;

theorem Th6: :: GRFUNC_1:6
for f being Function
for G being set st G c= f holds
G is Function
proof end;

theorem Th7: :: GRFUNC_1:7
canceled;

theorem Th8: :: GRFUNC_1:8
for f, g being Function holds
( f c= g iff ( dom f c= dom g & ( for x being set st x in dom f holds
f . x = g . x ) ) )
proof end;

theorem Th9: :: GRFUNC_1:9
for f, g being Function st dom f = dom g & f c= g holds
f = g
proof end;

Lemma23: for x, y being set
for f, h being Function st (rng f) /\ (rng h) = {} & x in dom f & y in dom h holds
f . x <> h . y
proof end;

theorem Th10: :: GRFUNC_1:10
canceled;

theorem Th11: :: GRFUNC_1:11
canceled;

theorem Th12: :: GRFUNC_1:12
for x, z being set
for g, f being Function st [x,z] in g * f holds
( [x,(f . x)] in f & [(f . x),z] in g )
proof end;

theorem Th13: :: GRFUNC_1:13
for h, f, g being Function st h c= f holds
( g * h c= g * f & h * g c= f * g ) by RELAT_1:48, RELAT_1:49;

theorem Th14: :: GRFUNC_1:14
canceled;

theorem Th15: :: GRFUNC_1:15
for x, y being set holds {[x,y]} is Function
proof end;

Lemma25: for x, y, x1, y1 being set st [x,y] in {[x1,y1]} holds
( x = x1 & y = y1 )
proof end;

theorem Th16: :: GRFUNC_1:16
for x, y being set
for f being Function st f = {[x,y]} holds
f . x = y
proof end;

theorem Th17: :: GRFUNC_1:17
canceled;

theorem Th18: :: GRFUNC_1:18
for x being set
for f being Function st dom f = {x} holds
f = {[x,(f . x)]}
proof end;

theorem Th19: :: GRFUNC_1:19
for x1, y1, x2, y2 being set holds
( {[x1,y1],[x2,y2]} is Function iff ( x1 = x2 implies y1 = y2 ) )
proof end;

theorem Th20: :: GRFUNC_1:20
canceled;

theorem Th21: :: GRFUNC_1:21
canceled;

theorem Th22: :: GRFUNC_1:22
canceled;

theorem Th23: :: GRFUNC_1:23
canceled;

theorem Th24: :: GRFUNC_1:24
canceled;

theorem Th25: :: GRFUNC_1:25
for f being Function holds
( f is one-to-one iff for x1, x2, y being set st [x1,y] in f & [x2,y] in f holds
x1 = x2 )
proof end;

theorem Th26: :: GRFUNC_1:26
for g, f being Function st g c= f & f is one-to-one holds
g is one-to-one
proof end;

theorem Th27: :: GRFUNC_1:27
for X being set
for f being Function holds f /\ X is Function
proof end;

theorem Th28: :: GRFUNC_1:28
for h, f, g being Function st h = f /\ g holds
( dom h c= (dom f) /\ (dom g) & rng h c= (rng f) /\ (rng g) ) by RELAT_1:14, RELAT_1:27;

theorem Th29: :: GRFUNC_1:29
for x being set
for h, f, g being Function st h = f /\ g & x in dom h holds
( h . x = f . x & h . x = g . x )
proof end;

theorem Th30: :: GRFUNC_1:30
for f, g, h being Function st ( f is one-to-one or g is one-to-one ) & h = f /\ g holds
h is one-to-one
proof end;

theorem Th31: :: GRFUNC_1:31
for f, g being Function st dom f misses dom g holds
f \/ g is Function
proof end;

theorem Th32: :: GRFUNC_1:32
for f, h, g being Function st f c= h & g c= h holds
f \/ g is Function
proof end;

Lemma31: for x being set
for h, f, g being Function st h = f \/ g holds
( x in dom h iff ( x in dom f or x in dom g ) )
proof end;

theorem Th33: :: GRFUNC_1:33
for h, f, g being Function st h = f \/ g holds
( dom h = (dom f) \/ (dom g) & rng h = (rng f) \/ (rng g) ) by RELAT_1:13, RELAT_1:26;

theorem Th34: :: GRFUNC_1:34
for x being set
for f, h, g being Function st x in dom f & h = f \/ g holds
h . x = f . x
proof end;

theorem Th35: :: GRFUNC_1:35
for x being set
for g, h, f being Function st x in dom g & h = f \/ g holds
h . x = g . x
proof end;

theorem Th36: :: GRFUNC_1:36
for x being set
for h, f, g being Function st x in dom h & h = f \/ g & not h . x = f . x holds
h . x = g . x
proof end;

theorem Th37: :: GRFUNC_1:37
for f, g, h being Function st f is one-to-one & g is one-to-one & h = f \/ g & rng f misses rng g holds
h is one-to-one
proof end;

theorem Th38: :: GRFUNC_1:38
for X being set
for f being Function holds f \ X is Function
proof end;

theorem Th39: :: GRFUNC_1:39
canceled;

theorem Th40: :: GRFUNC_1:40
canceled;

theorem Th41: :: GRFUNC_1:41
canceled;

theorem Th42: :: GRFUNC_1:42
canceled;

theorem Th43: :: GRFUNC_1:43
canceled;

theorem Th44: :: GRFUNC_1:44
canceled;

theorem Th45: :: GRFUNC_1:45
canceled;

theorem Th46: :: GRFUNC_1:46
for f being Function st f = {} holds
f is one-to-one
proof end;

theorem Th47: :: GRFUNC_1:47
for f being Function st f is one-to-one holds
for x, y being set holds
( [y,x] in f " iff [x,y] in f )
proof end;

theorem Th48: :: GRFUNC_1:48
canceled;

theorem Th49: :: GRFUNC_1:49
for f being Function st f = {} holds
f " = {}
proof end;

theorem Th50: :: GRFUNC_1:50
canceled;

theorem Th51: :: GRFUNC_1:51
canceled;

theorem Th52: :: GRFUNC_1:52
for x, X being set
for f being Function holds
( ( x in dom f & x in X ) iff [x,(f . x)] in f | X )
proof end;

theorem Th53: :: GRFUNC_1:53
canceled;

theorem Th54: :: GRFUNC_1:54
canceled;

theorem Th55: :: GRFUNC_1:55
canceled;

theorem Th56: :: GRFUNC_1:56
canceled;

theorem Th57: :: GRFUNC_1:57
canceled;

theorem Th58: :: GRFUNC_1:58
canceled;

theorem Th59: :: GRFUNC_1:59
canceled;

theorem Th60: :: GRFUNC_1:60
canceled;

theorem Th61: :: GRFUNC_1:61
canceled;

theorem Th62: :: GRFUNC_1:62
canceled;

theorem Th63: :: GRFUNC_1:63
canceled;

theorem Th64: :: GRFUNC_1:64
for g, f being Function st g c= f holds
f | (dom g) = g
proof end;

theorem Th65: :: GRFUNC_1:65
canceled;

theorem Th66: :: GRFUNC_1:66
canceled;

theorem Th67: :: GRFUNC_1:67
for x, Y being set
for f being Function holds
( ( x in dom f & f . x in Y ) iff [x,(f . x)] in Y | f )
proof end;

theorem Th68: :: GRFUNC_1:68
canceled;

theorem Th69: :: GRFUNC_1:69
canceled;

theorem Th70: :: GRFUNC_1:70
canceled;

theorem Th71: :: GRFUNC_1:71
canceled;

theorem Th72: :: GRFUNC_1:72
canceled;

theorem Th73: :: GRFUNC_1:73
canceled;

theorem Th74: :: GRFUNC_1:74
canceled;

theorem Th75: :: GRFUNC_1:75
canceled;

theorem Th76: :: GRFUNC_1:76
canceled;

theorem Th77: :: GRFUNC_1:77
canceled;

theorem Th78: :: GRFUNC_1:78
canceled;

theorem Th79: :: GRFUNC_1:79
for g, f being Function st g c= f & f is one-to-one holds
(rng g) | f = g
proof end;

theorem Th80: :: GRFUNC_1:80
canceled;

theorem Th81: :: GRFUNC_1:81
canceled;

theorem Th82: :: GRFUNC_1:82
canceled;

theorem Th83: :: GRFUNC_1:83
canceled;

theorem Th84: :: GRFUNC_1:84
canceled;

theorem Th85: :: GRFUNC_1:85
canceled;

theorem Th86: :: GRFUNC_1:86
canceled;

theorem Th87: :: GRFUNC_1:87
for x, Y being set
for f being Function holds
( x in f " Y iff ( [x,(f . x)] in f & f . x in Y ) )
proof end;

theorem Th88: :: GRFUNC_1:88
for X being set
for f, g being Function st X c= dom f & f c= g holds
f | X = g | X
proof end;

theorem Th89: :: GRFUNC_1:89
for f being Function
for x being set st x in dom f holds
f | {x} = {[x,(f . x)]}
proof end;