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Mirrors > Home > ILE Home > Th. List > coi1 | Unicode version |
Description: Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
Ref | Expression |
---|---|
coi1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relco 5096 | . 2 | |
2 | vex 2724 | . . . . . 6 | |
3 | vex 2724 | . . . . . 6 | |
4 | 2, 3 | opelco 4770 | . . . . 5 |
5 | vex 2724 | . . . . . . . . . 10 | |
6 | 5 | ideq 4750 | . . . . . . . . 9 |
7 | equcom 1693 | . . . . . . . . 9 | |
8 | 6, 7 | bitri 183 | . . . . . . . 8 |
9 | 8 | anbi1i 454 | . . . . . . 7 |
10 | 9 | exbii 1592 | . . . . . 6 |
11 | breq1 3979 | . . . . . . 7 | |
12 | 2, 11 | ceqsexv 2760 | . . . . . 6 |
13 | 10, 12 | bitri 183 | . . . . 5 |
14 | 4, 13 | bitri 183 | . . . 4 |
15 | df-br 3977 | . . . 4 | |
16 | 14, 15 | bitri 183 | . . 3 |
17 | 16 | eqrelriv 4691 | . 2 |
18 | 1, 17 | mpan 421 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wex 1479 wcel 2135 cop 3573 class class class wbr 3976 cid 4260 ccom 4602 wrel 4603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-co 4607 |
This theorem is referenced by: coi2 5114 coires1 5115 relcoi1 5129 fcoi1 5362 |
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