:: FUNCT_2 semantic presentation
:: deftheorem Def1 defines quasi_total FUNCT_2:def 1 :
theorem Th1: :: FUNCT_2:1
canceled;
theorem Th2: :: FUNCT_2:2
canceled;
theorem Th3: :: FUNCT_2:3
theorem Th4: :: FUNCT_2:4
theorem Th5: :: FUNCT_2:5
theorem Th6: :: FUNCT_2:6
theorem Th7: :: FUNCT_2:7
theorem Th8: :: FUNCT_2:8
theorem Th9: :: FUNCT_2:9
:: deftheorem Def2 defines Funcs FUNCT_2:def 2 :
theorem Th10: :: FUNCT_2:10
canceled;
theorem Th11: :: FUNCT_2:11
theorem Th12: :: FUNCT_2:12
theorem Th13: :: FUNCT_2:13
canceled;
theorem Th14: :: FUNCT_2:14
theorem Th15: :: FUNCT_2:15
canceled;
theorem Th16: :: FUNCT_2:16
theorem Th17: :: FUNCT_2:17
theorem Th18: :: FUNCT_2:18
for
X,
Y being
set for
f1,
f2 being
Function of
X,
Y st ( for
x being
set st
x in X holds
f1 . x = f2 . x ) holds
f1 = f2
theorem Th19: :: FUNCT_2:19
theorem Th20: :: FUNCT_2:20
theorem Th21: :: FUNCT_2:21
theorem Th22: :: FUNCT_2:22
theorem Th23: :: FUNCT_2:23
theorem Th24: :: FUNCT_2:24
theorem Th25: :: FUNCT_2:25
theorem Th26: :: FUNCT_2:26
theorem Th27: :: FUNCT_2:27
theorem Th28: :: FUNCT_2:28
theorem Th29: :: FUNCT_2:29
theorem Th30: :: FUNCT_2:30
theorem Th31: :: FUNCT_2:31
theorem Th32: :: FUNCT_2:32
theorem Th33: :: FUNCT_2:33
canceled;
theorem Th34: :: FUNCT_2:34
theorem Th35: :: FUNCT_2:35
theorem Th36: :: FUNCT_2:36
theorem Th37: :: FUNCT_2:37
theorem Th38: :: FUNCT_2:38
theorem Th39: :: FUNCT_2:39
canceled;
theorem Th40: :: FUNCT_2:40
for
X,
Y,
Z being
set for
f being
Function of
X,
Y st
X c= Z holds
f | Z = f
theorem Th41: :: FUNCT_2:41
theorem Th42: :: FUNCT_2:42
canceled;
theorem Th43: :: FUNCT_2:43
theorem Th44: :: FUNCT_2:44
theorem Th45: :: FUNCT_2:45
theorem Th46: :: FUNCT_2:46
for
X,
Y,
Q being
set for
f being
Function of
X,
Y st
Y <> {} holds
for
x being
set holds
(
x in f " Q iff (
x in X &
f . x in Q ) )
theorem Th47: :: FUNCT_2:47
theorem Th48: :: FUNCT_2:48
theorem Th49: :: FUNCT_2:49
theorem Th50: :: FUNCT_2:50
theorem Th51: :: FUNCT_2:51
theorem Th52: :: FUNCT_2:52
canceled;
theorem Th53: :: FUNCT_2:53
theorem Th54: :: FUNCT_2:54
canceled;
theorem Th55: :: FUNCT_2:55
theorem Th56: :: FUNCT_2:56
canceled;
theorem Th57: :: FUNCT_2:57
canceled;
theorem Th58: :: FUNCT_2:58
canceled;
theorem Th59: :: FUNCT_2:59
theorem Th60: :: FUNCT_2:60
theorem Th61: :: FUNCT_2:61
theorem Th62: :: FUNCT_2:62
theorem Th63: :: FUNCT_2:63
canceled;
theorem Th64: :: FUNCT_2:64
theorem Th65: :: FUNCT_2:65
theorem Th66: :: FUNCT_2:66
theorem Th67: :: FUNCT_2:67
theorem Th68: :: FUNCT_2:68
canceled;
theorem Th69: :: FUNCT_2:69
canceled;
theorem Th70: :: FUNCT_2:70
theorem Th71: :: FUNCT_2:71
canceled;
theorem Th72: :: FUNCT_2:72
canceled;
theorem Th73: :: FUNCT_2:73
theorem Th74: :: FUNCT_2:74
canceled;
theorem Th75: :: FUNCT_2:75
theorem Th76: :: FUNCT_2:76
theorem Th77: :: FUNCT_2:77
theorem Th78: :: FUNCT_2:78
canceled;
theorem Th79: :: FUNCT_2:79
theorem Th80: :: FUNCT_2:80
canceled;
theorem Th81: :: FUNCT_2:81
canceled;
theorem Th82: :: FUNCT_2:82
:: deftheorem Def3 defines onto FUNCT_2:def 3 :
:: deftheorem Def4 defines bijective FUNCT_2:def 4 :
theorem Th83: :: FUNCT_2:83
theorem Th84: :: FUNCT_2:84
canceled;
theorem Th85: :: FUNCT_2:85
theorem Th86: :: FUNCT_2:86
theorem Th87: :: FUNCT_2:87
theorem Th88: :: FUNCT_2:88
theorem Th89: :: FUNCT_2:89
canceled;
theorem Th90: :: FUNCT_2:90
canceled;
theorem Th91: :: FUNCT_2:91
canceled;
theorem Th92: :: FUNCT_2:92
theorem Th93: :: FUNCT_2:93
canceled;
theorem Th94: :: FUNCT_2:94
canceled;
theorem Th95: :: FUNCT_2:95
canceled;
theorem Th96: :: FUNCT_2:96
canceled;
theorem Th97: :: FUNCT_2:97
canceled;
theorem Th98: :: FUNCT_2:98
canceled;
theorem Th99: :: FUNCT_2:99
canceled;
theorem Th100: :: FUNCT_2:100
canceled;
theorem Th101: :: FUNCT_2:101
canceled;
theorem Th102: :: FUNCT_2:102
canceled;
theorem Th103: :: FUNCT_2:103
canceled;
theorem Th104: :: FUNCT_2:104
canceled;
theorem Th105: :: FUNCT_2:105
canceled;
theorem Th106: :: FUNCT_2:106
canceled;
theorem Th107: :: FUNCT_2:107
canceled;
theorem Th108: :: FUNCT_2:108
canceled;
theorem Th109: :: FUNCT_2:109
canceled;
theorem Th110: :: FUNCT_2:110
canceled;
theorem Th111: :: FUNCT_2:111
canceled;
theorem Th112: :: FUNCT_2:112
canceled;
theorem Th113: :: FUNCT_2:113
for
X,
Y being
set for
f1,
f2 being
Function of
X,
Y st ( for
x being
Element of
X holds
f1 . x = f2 . x ) holds
f1 = f2
theorem Th114: :: FUNCT_2:114
canceled;
theorem Th115: :: FUNCT_2:115
for
X,
Y,
P being
set for
f being
Function of
X,
Y for
y being
set st
y in f .: P holds
ex
x being
set st
(
x in X &
x in P &
y = f . x )
theorem Th116: :: FUNCT_2:116
theorem Th117: :: FUNCT_2:117
canceled;
theorem Th118: :: FUNCT_2:118
canceled;
theorem Th119: :: FUNCT_2:119
canceled;
theorem Th120: :: FUNCT_2:120
canceled;
theorem Th121: :: FUNCT_2:121
scheme :: FUNCT_2:sch 82
s82{
F1()
-> set ,
F2()
-> set ,
P1[
set ],
F3(
set )
-> set ,
F4(
set )
-> set } :
ex
f being
Function of
F1(),
F2() st
for
x being
set st
x in F1() holds
( (
P1[
x] implies
f . x = F3(
x) ) & (
P1[
x] implies
f . x = F4(
x) ) )
provided
E20:
for
x being
set st
x in F1() holds
( (
P1[
x] implies
F3(
x)
in F2() ) & (
P1[
x] implies
F4(
x)
in F2() ) )
theorem Th122: :: FUNCT_2:122
canceled;
theorem Th123: :: FUNCT_2:123
canceled;
theorem Th124: :: FUNCT_2:124
canceled;
theorem Th125: :: FUNCT_2:125
canceled;
theorem Th126: :: FUNCT_2:126
canceled;
theorem Th127: :: FUNCT_2:127
canceled;
theorem Th128: :: FUNCT_2:128
canceled;
theorem Th129: :: FUNCT_2:129
canceled;
theorem Th130: :: FUNCT_2:130
theorem Th131: :: FUNCT_2:131
theorem Th132: :: FUNCT_2:132
theorem Th133: :: FUNCT_2:133
theorem Th134: :: FUNCT_2:134
theorem Th135: :: FUNCT_2:135
canceled;
theorem Th136: :: FUNCT_2:136
theorem Th137: :: FUNCT_2:137
canceled;
theorem Th138: :: FUNCT_2:138
canceled;
theorem Th139: :: FUNCT_2:139
canceled;
theorem Th140: :: FUNCT_2:140
canceled;
theorem Th141: :: FUNCT_2:141
theorem Th142: :: FUNCT_2:142
theorem Th143: :: FUNCT_2:143
theorem Th144: :: FUNCT_2:144
canceled;
theorem Th145: :: FUNCT_2:145
theorem Th146: :: FUNCT_2:146
theorem Th147: :: FUNCT_2:147
canceled;
theorem Th148: :: FUNCT_2:148
theorem Th149: :: FUNCT_2:149
theorem Th150: :: FUNCT_2:150
canceled;
theorem Th151: :: FUNCT_2:151
theorem Th152: :: FUNCT_2:152
theorem Th153: :: FUNCT_2:153
canceled;
theorem Th154: :: FUNCT_2:154
theorem Th155: :: FUNCT_2:155
theorem Th156: :: FUNCT_2:156
theorem Th157: :: FUNCT_2:157
canceled;
theorem Th158: :: FUNCT_2:158
theorem Th159: :: FUNCT_2:159
theorem Th160: :: FUNCT_2:160
theorem Th161: :: FUNCT_2:161
theorem Th162: :: FUNCT_2:162
theorem Th163: :: FUNCT_2:163
canceled;
theorem Th164: :: FUNCT_2:164
theorem Th165: :: FUNCT_2:165
theorem Th166: :: FUNCT_2:166
theorem Th167: :: FUNCT_2:167
theorem Th168: :: FUNCT_2:168
definition
let o be
set ,
m be
set ,
r be
set ;
func c1,
c2 :-> c3 -> Function of
[:{a1},{a2}:],
{a3} means :: FUNCT_2:def 5
verum;
existence
ex b1 being Function of [:{o},{m}:],{r} st verum
;
uniqueness
for b1, b2 being Function of [:{o},{m}:],{r} holds b1 = b2
by ;
end;
:: deftheorem Def5 defines :-> FUNCT_2:def 5 :
theorem Th169: :: FUNCT_2:169
definition
let A be non
empty set ,
B be non
empty set ,
C be non
empty set ;
let f be
Function of
A,
[:B,C:];
redefine func pr1 as
pr1 c4 -> Function of
a1,
a2 means :: FUNCT_2:def 6
for
x being
Element of
A holds
it . x = (f . x) `1 ;
coherence
pr1 f is Function of A,B
compatibility
for b1 being Function of A,B holds
( b1 = pr1 f iff for x being Element of A holds b1 . x = (f . x) `1 )
redefine func pr2 as
pr2 c4 -> Function of
a1,
a3 means :: FUNCT_2:def 7
for
x being
Element of
A holds
it . x = (f . x) `2 ;
coherence
pr2 f is Function of A,C
compatibility
for b1 being Function of A,C holds
( b1 = pr2 f iff for x being Element of A holds b1 . x = (f . x) `2 )
end;
:: deftheorem Def6 defines pr1 FUNCT_2:def 6 :
:: deftheorem Def7 defines pr2 FUNCT_2:def 7 :
:: deftheorem Def8 defines = FUNCT_2:def 8 :
:: deftheorem Def9 defines = FUNCT_2:def 9 :
theorem Th170: :: FUNCT_2:170