:: Functions and Their Basic Properties
:: by Czes{\l}aw Byli\'nski
::
:: Received March 3, 1989
:: Copyright (c) 1990 Association of Mizar Users
:: deftheorem Def1 defines Function-like FUNCT_1:def 1 :
theorem :: FUNCT_1:1
canceled;
theorem :: FUNCT_1:2
canceled;
:: 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
theorem Th9: :: FUNCT_1:9
:: deftheorem Def5 defines rng FUNCT_1:def 5 :
theorem :: FUNCT_1:10
canceled;
theorem :: FUNCT_1:11
canceled;
theorem :: FUNCT_1:12
theorem :: FUNCT_1:13
canceled;
theorem Th14: :: FUNCT_1:14
theorem Th15: :: FUNCT_1:15
theorem :: FUNCT_1:16
theorem :: FUNCT_1:17
theorem :: FUNCT_1:18
theorem :: FUNCT_1:19
theorem :: FUNCT_1:20
theorem Th21: :: FUNCT_1:21
theorem Th22: :: FUNCT_1:22
theorem Th23: :: FUNCT_1:23
theorem :: FUNCT_1:24
canceled;
theorem :: FUNCT_1:25
theorem :: FUNCT_1:26
canceled;
theorem Th27: :: FUNCT_1:27
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
theorem Th34: :: FUNCT_1:34
theorem :: FUNCT_1:35
theorem :: FUNCT_1:36
canceled;
theorem Th37: :: FUNCT_1:37
theorem :: FUNCT_1:38
theorem :: FUNCT_1:39
canceled;
theorem :: FUNCT_1:40
theorem :: FUNCT_1:41
canceled;
theorem :: FUNCT_1:42
canceled;
theorem :: FUNCT_1:43
theorem Th44: :: FUNCT_1:44
:: deftheorem FUNCT_1:def 6 :
canceled;
:: deftheorem FUNCT_1:def 7 :
canceled;
:: deftheorem Def8 defines one-to-one FUNCT_1:def 8 :
theorem :: FUNCT_1:45
canceled;
theorem Th46: :: FUNCT_1:46
theorem Th47: :: FUNCT_1:47
theorem :: FUNCT_1:48
theorem :: FUNCT_1:49
theorem :: FUNCT_1:50
theorem :: FUNCT_1:51
theorem Th52: :: FUNCT_1:52
theorem :: 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 :: FUNCT_1:60
theorem Th61: :: FUNCT_1:61
theorem Th62: :: FUNCT_1:62
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
theorem Th63: :: FUNCT_1:63
theorem :: FUNCT_1:64
theorem :: FUNCT_1:65
theorem :: FUNCT_1:66
theorem :: FUNCT_1:67
theorem Th68: :: FUNCT_1:68
theorem :: FUNCT_1:69
canceled;
theorem Th70: :: FUNCT_1:70
theorem Th71: :: FUNCT_1:71
theorem Th72: :: FUNCT_1:72
theorem :: FUNCT_1:73
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
theorem :: FUNCT_1:83
canceled;
theorem :: FUNCT_1:84
theorem Th85: :: FUNCT_1:85
theorem :: FUNCT_1:86
theorem :: FUNCT_1:87
theorem :: FUNCT_1:88
canceled;
theorem :: FUNCT_1:89
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
theorem :: FUNCT_1:98
canceled;
theorem :: FUNCT_1:99
:: 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
theorem :: FUNCT_1:118
theorem :: FUNCT_1:119
canceled;
theorem :: FUNCT_1:120
theorem Th121: :: FUNCT_1:121
theorem :: FUNCT_1:122
theorem :: FUNCT_1:123
theorem :: FUNCT_1:124
theorem :: FUNCT_1:125
theorem :: 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 :: 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
theorem :: FUNCT_1:138
theorem :: FUNCT_1:139
theorem :: FUNCT_1:140
canceled;
theorem :: FUNCT_1:141
theorem Th142: :: FUNCT_1:142
theorem :: FUNCT_1:143
theorem Th144: :: FUNCT_1:144
theorem Th145: :: FUNCT_1:145
theorem Th146: :: FUNCT_1:146
theorem :: FUNCT_1:147
theorem :: FUNCT_1:148
theorem Th149: :: FUNCT_1:149
theorem :: FUNCT_1:150
theorem :: FUNCT_1:151
theorem Th152: :: FUNCT_1:152
theorem :: FUNCT_1:153
theorem :: FUNCT_1:154
theorem :: FUNCT_1:155
theorem :: FUNCT_1:156
theorem :: FUNCT_1:157
theorem Th158: :: FUNCT_1:158
theorem :: FUNCT_1:159
theorem :: FUNCT_1:160
theorem :: FUNCT_1:161
theorem :: FUNCT_1:162
:: deftheorem 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 :: FUNCT_1:163
theorem :: FUNCT_1:164
:: deftheorem defines = FUNCT_1:def 17 :
theorem :: FUNCT_1:165
theorem :: FUNCT_1:166
theorem Th167: :: FUNCT_1:167
theorem :: FUNCT_1:168
theorem :: FUNCT_1:169
theorem :: FUNCT_1:170
theorem :: FUNCT_1:171