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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 4819 | . 2 | |
2 | vex 2560 | . . . . . 6 | |
3 | vex 2560 | . . . . . 6 | |
4 | 2, 3 | opelco 4507 | . . . . 5 |
5 | vex 2560 | . . . . . . . . . 10 | |
6 | 5 | ideq 4488 | . . . . . . . . 9 |
7 | equcom 1593 | . . . . . . . . 9 | |
8 | 6, 7 | bitri 173 | . . . . . . . 8 |
9 | 8 | anbi1i 431 | . . . . . . 7 |
10 | 9 | exbii 1496 | . . . . . 6 |
11 | breq1 3767 | . . . . . . 7 | |
12 | 2, 11 | ceqsexv 2593 | . . . . . 6 |
13 | 10, 12 | bitri 173 | . . . . 5 |
14 | 4, 13 | bitri 173 | . . . 4 |
15 | df-br 3765 | . . . 4 | |
16 | 14, 15 | bitri 173 | . . 3 |
17 | 16 | eqrelriv 4433 | . 2 |
18 | 1, 17 | mpan 400 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wex 1381 wcel 1393 cop 3378 class class class wbr 3764 cid 4025 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-co 4354 |
This theorem is referenced by: coi2 4837 coires1 4838 relcoi1 4849 fcoi1 5070 |
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