:: FUNCT_1 semantic presentation
:: deftheorem Def1 defines Function-like FUNCT_1:def 1 :
theorem Th1: :: FUNCT_1:1
canceled;
theorem Th2: :: FUNCT_1:2
:: deftheorem Def2 FUNCT_1:def 2 :
canceled;
:: deftheorem Def3 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 Th3: :: FUNCT_1:3
canceled;
theorem Th4: :: FUNCT_1:4
canceled;
theorem Th5: :: FUNCT_1:5
canceled;
theorem Th6: :: FUNCT_1:6
canceled;
theorem Th7: :: FUNCT_1:7
canceled;
theorem Th8: :: FUNCT_1:8
theorem Th9: :: FUNCT_1:9
:: deftheorem Def5 defines rng FUNCT_1:def 5 :
theorem Th10: :: FUNCT_1:10
canceled;
theorem Th11: :: FUNCT_1:11
canceled;
theorem Th12: :: FUNCT_1:12
theorem Th13: :: FUNCT_1:13
canceled;
theorem Th14: :: FUNCT_1:14
theorem Th15: :: FUNCT_1:15
theorem Th16: :: FUNCT_1:16
theorem Th17: :: FUNCT_1:17
theorem Th18: :: FUNCT_1:18
theorem Th19: :: FUNCT_1:19
theorem Th20: :: FUNCT_1:20
theorem Th21: :: FUNCT_1:21
theorem Th22: :: FUNCT_1:22
theorem Th23: :: FUNCT_1:23
theorem Th24: :: FUNCT_1:24
canceled;
theorem Th25: :: FUNCT_1:25
theorem Th26: :: FUNCT_1:26
canceled;
theorem Th27: :: FUNCT_1:27
theorem Th28: :: FUNCT_1:28
canceled;
theorem Th29: :: FUNCT_1:29
canceled;
theorem Th30: :: FUNCT_1:30
canceled;
theorem Th31: :: FUNCT_1:31
canceled;
theorem Th32: :: FUNCT_1:32
canceled;
theorem Th33: :: FUNCT_1:33
theorem Th34: :: FUNCT_1:34
theorem Th35: :: FUNCT_1:35
for
x,
X being
set st
x in X holds
(id X) . x = x by ;
theorem Th36: :: FUNCT_1:36
canceled;
theorem Th37: :: FUNCT_1:37
theorem Th38: :: FUNCT_1:38
theorem Th39: :: FUNCT_1:39
canceled;
theorem Th40: :: FUNCT_1:40
theorem Th41: :: FUNCT_1:41
canceled;
theorem Th42: :: FUNCT_1:42
canceled;
theorem Th43: :: FUNCT_1:43
theorem Th44: :: FUNCT_1:44
:: deftheorem Def6 FUNCT_1:def 6 :
canceled;
:: deftheorem Def7 FUNCT_1:def 7 :
canceled;
:: deftheorem Def8 defines one-to-one FUNCT_1:def 8 :
theorem Th45: :: FUNCT_1:45
canceled;
theorem Th46: :: FUNCT_1:46
theorem Th47: :: FUNCT_1:47
theorem Th48: :: FUNCT_1:48
theorem Th49: :: FUNCT_1:49
theorem Th50: :: FUNCT_1:50
theorem Th51: :: FUNCT_1:51
theorem Th52: :: FUNCT_1:52
theorem Th53: :: FUNCT_1:53
:: deftheorem Def9 defines " FUNCT_1:def 9 :
theorem Th54: :: FUNCT_1:54
theorem Th55: :: FUNCT_1:55
theorem Th56: :: FUNCT_1:56
theorem Th57: :: FUNCT_1:57
theorem Th58: :: FUNCT_1:58
theorem Th59: :: FUNCT_1:59
theorem Th60: :: FUNCT_1:60
theorem Th61: :: FUNCT_1:61
theorem Th62: :: FUNCT_1:62
Lemma82:
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
theorem Th63: :: FUNCT_1:63
theorem Th64: :: FUNCT_1:64
theorem Th65: :: FUNCT_1:65
theorem Th66: :: FUNCT_1:66
theorem Th67: :: FUNCT_1:67
theorem Th68: :: FUNCT_1:68
theorem Th69: :: FUNCT_1:69
canceled;
theorem Th70: :: FUNCT_1:70
theorem Th71: :: FUNCT_1:71
theorem Th72: :: FUNCT_1:72
theorem Th73: :: FUNCT_1:73
theorem Th74: :: FUNCT_1:74
canceled;
theorem Th75: :: FUNCT_1:75
canceled;
theorem Th76: :: FUNCT_1:76
canceled;
theorem Th77: :: FUNCT_1:77
canceled;
theorem Th78: :: FUNCT_1:78
canceled;
theorem Th79: :: FUNCT_1:79
canceled;
theorem Th80: :: FUNCT_1:80
canceled;
theorem Th81: :: FUNCT_1:81
canceled;
theorem Th82: :: FUNCT_1:82
theorem Th83: :: FUNCT_1:83
canceled;
theorem Th84: :: FUNCT_1:84
theorem Th85: :: FUNCT_1:85
theorem Th86: :: FUNCT_1:86
theorem Th87: :: FUNCT_1:87
theorem Th88: :: FUNCT_1:88
canceled;
theorem Th89: :: FUNCT_1:89
theorem Th90: :: FUNCT_1:90
canceled;
theorem Th91: :: FUNCT_1:91
canceled;
theorem Th92: :: FUNCT_1:92
canceled;
theorem Th93: :: FUNCT_1:93
canceled;
theorem Th94: :: FUNCT_1:94
canceled;
theorem Th95: :: FUNCT_1:95
canceled;
theorem Th96: :: FUNCT_1:96
canceled;
theorem Th97: :: FUNCT_1:97
theorem Th98: :: FUNCT_1:98
canceled;
theorem Th99: :: FUNCT_1:99
:: deftheorem Def10 FUNCT_1:def 10 :
canceled;
:: deftheorem Def11 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 Th100: :: FUNCT_1:100
canceled;
theorem Th101: :: FUNCT_1:101
canceled;
theorem Th102: :: FUNCT_1:102
canceled;
theorem Th103: :: FUNCT_1:103
canceled;
theorem Th104: :: FUNCT_1:104
canceled;
theorem Th105: :: FUNCT_1:105
canceled;
theorem Th106: :: FUNCT_1:106
canceled;
theorem Th107: :: FUNCT_1:107
canceled;
theorem Th108: :: FUNCT_1:108
canceled;
theorem Th109: :: FUNCT_1:109
canceled;
theorem Th110: :: FUNCT_1:110
canceled;
theorem Th111: :: FUNCT_1:111
canceled;
theorem Th112: :: FUNCT_1:112
canceled;
theorem Th113: :: FUNCT_1:113
canceled;
theorem Th114: :: FUNCT_1:114
canceled;
theorem Th115: :: FUNCT_1:115
canceled;
theorem Th116: :: FUNCT_1:116
canceled;
theorem Th117: :: FUNCT_1:117
theorem Th118: :: FUNCT_1:118
theorem Th119: :: FUNCT_1:119
canceled;
theorem Th120: :: FUNCT_1:120
theorem Th121: :: FUNCT_1:121
theorem Th122: :: FUNCT_1:122
theorem Th123: :: FUNCT_1:123
theorem Th124: :: FUNCT_1:124
theorem Th125: :: FUNCT_1:125
theorem Th126: :: FUNCT_1:126
:: 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 Th127: :: FUNCT_1:127
canceled;
theorem Th128: :: FUNCT_1:128
canceled;
theorem Th129: :: FUNCT_1:129
canceled;
theorem Th130: :: FUNCT_1:130
canceled;
theorem Th131: :: FUNCT_1:131
canceled;
theorem Th132: :: FUNCT_1:132
canceled;
theorem Th133: :: FUNCT_1:133
canceled;
theorem Th134: :: FUNCT_1:134
canceled;
theorem Th135: :: FUNCT_1:135
canceled;
theorem Th136: :: FUNCT_1:136
canceled;
theorem Th137: :: FUNCT_1:137
theorem Th138: :: FUNCT_1:138
theorem Th139: :: FUNCT_1:139
theorem Th140: :: FUNCT_1:140
canceled;
theorem Th141: :: FUNCT_1:141
theorem Th142: :: FUNCT_1:142
theorem Th143: :: FUNCT_1:143
theorem Th144: :: FUNCT_1:144
theorem Th145: :: FUNCT_1:145
theorem Th146: :: FUNCT_1:146
theorem Th147: :: FUNCT_1:147
theorem Th148: :: FUNCT_1:148
theorem Th149: :: FUNCT_1:149
theorem Th150: :: FUNCT_1:150
theorem Th151: :: FUNCT_1:151
theorem Th152: :: FUNCT_1:152
theorem Th153: :: FUNCT_1:153
theorem Th154: :: FUNCT_1:154
theorem Th155: :: FUNCT_1:155
theorem Th156: :: FUNCT_1:156
theorem Th157: :: FUNCT_1:157
theorem Th158: :: FUNCT_1:158
theorem Th159: :: FUNCT_1:159
theorem Th160: :: FUNCT_1:160
theorem Th161: :: FUNCT_1:161
theorem Th162: :: FUNCT_1:162
:: deftheorem Def14 defines empty-yielding FUNCT_1:def 14 :
:: deftheorem Def15 defines non-empty FUNCT_1:def 15 :
:: deftheorem Def16 defines constant FUNCT_1:def 16 :
theorem Th163: :: FUNCT_1:163
theorem Th164: :: FUNCT_1:164