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Mirrors > Home > ILE Home > Th. List > iss | Unicode version |
Description: A subclass of the identity function is the identity function restricted to its domain. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
iss |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3149 |
. . . . . . 7
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2 | vex 2740 |
. . . . . . . . 9
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3 | vex 2740 |
. . . . . . . . 9
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4 | 2, 3 | opeldm 4826 |
. . . . . . . 8
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5 | 4 | a1i 9 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 1, 5 | jcad 307 |
. . . . . 6
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7 | df-br 4001 |
. . . . . . . . 9
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8 | 3 | ideq 4775 |
. . . . . . . . 9
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9 | 7, 8 | bitr3i 186 |
. . . . . . . 8
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10 | 2 | eldm2 4821 |
. . . . . . . . . 10
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11 | opeq2 3777 |
. . . . . . . . . . . . . . 15
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12 | 11 | eleq1d 2246 |
. . . . . . . . . . . . . 14
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13 | 12 | biimprcd 160 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 9, 13 | biimtrid 152 |
. . . . . . . . . . . 12
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15 | 1, 14 | sylcom 28 |
. . . . . . . . . . 11
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16 | 15 | exlimdv 1819 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 10, 16 | biimtrid 152 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 12 | imbi2d 230 |
. . . . . . . . 9
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19 | 17, 18 | syl5ibcom 155 |
. . . . . . . 8
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20 | 9, 19 | biimtrid 152 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 20 | impd 254 |
. . . . . 6
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22 | 6, 21 | impbid 129 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 3 | opelres 4908 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 22, 23 | bitr4di 198 |
. . . 4
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25 | 24 | alrimivv 1875 |
. . 3
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26 | reli 4752 |
. . . . 5
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27 | relss 4710 |
. . . . 5
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28 | 26, 27 | mpi 15 |
. . . 4
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29 | relres 4931 |
. . . 4
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30 | eqrel 4712 |
. . . 4
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31 | 28, 29, 30 | sylancl 413 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 25, 31 | mpbird 167 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | resss 4927 |
. . 3
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34 | sseq1 3178 |
. . 3
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35 | 33, 34 | mpbiri 168 |
. 2
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36 | 32, 35 | impbii 126 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-dm 4633 df-res 4635 |
This theorem is referenced by: funcocnv2 5482 |
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