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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 4520 | . . 3 | |
2 | relcnv 4703 | . . . . 5 | |
3 | coi1 4836 | . . . . 5 | |
4 | 2, 3 | ax-mp 7 | . . . 4 |
5 | 4 | cnveqi 4510 | . . 3 |
6 | 1, 5 | eqtr3i 2062 | . 2 |
7 | dfrel2 4771 | . . 3 | |
8 | cnvi 4728 | . . . 4 | |
9 | coeq2 4494 | . . . . 5 | |
10 | coeq1 4493 | . . . . 5 | |
11 | 9, 10 | sylan9eq 2092 | . . . 4 |
12 | 8, 11 | mpan2 401 | . . 3 |
13 | 7, 12 | sylbi 114 | . 2 |
14 | 7 | biimpi 113 | . 2 |
15 | 6, 13, 14 | 3eqtr3a 2096 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1243 cid 4025 ccnv 4344 ccom 4349 wrel 4350 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 |
This theorem is referenced by: relcoi2 4848 funi 4932 fcoi2 5071 |
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