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Mirrors > Home > ILE Home > Th. List > coi2 | 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 |
---|---|
coi2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvco 4538 | . . 3 | |
2 | relcnv 4723 | . . . . 5 | |
3 | coi1 4856 | . . . . 5 | |
4 | 2, 3 | ax-mp 7 | . . . 4 |
5 | 4 | cnveqi 4528 | . . 3 |
6 | 1, 5 | eqtr3i 2103 | . 2 |
7 | dfrel2 4791 | . . 3 | |
8 | cnvi 4748 | . . . 4 | |
9 | coeq2 4512 | . . . . 5 | |
10 | coeq1 4511 | . . . . 5 | |
11 | 9, 10 | sylan9eq 2133 | . . . 4 |
12 | 8, 11 | mpan2 415 | . . 3 |
13 | 7, 12 | sylbi 119 | . 2 |
14 | 7 | biimpi 118 | . 2 |
15 | 6, 13, 14 | 3eqtr3a 2137 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1284 cid 4043 ccnv 4362 ccom 4367 wrel 4368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 |
This theorem is referenced by: relcoi2 4868 funi 4952 fcoi2 5091 |
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