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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 4796 | . . 3 | |
2 | relcnv 4989 | . . . . 5 | |
3 | coi1 5126 | . . . . 5 | |
4 | 2, 3 | ax-mp 5 | . . . 4 |
5 | 4 | cnveqi 4786 | . . 3 |
6 | 1, 5 | eqtr3i 2193 | . 2 |
7 | dfrel2 5061 | . . 3 | |
8 | cnvi 5015 | . . . 4 | |
9 | coeq2 4769 | . . . . 5 | |
10 | coeq1 4768 | . . . . 5 | |
11 | 9, 10 | sylan9eq 2223 | . . . 4 |
12 | 8, 11 | mpan2 423 | . . 3 |
13 | 7, 12 | sylbi 120 | . 2 |
14 | 7 | biimpi 119 | . 2 |
15 | 6, 13, 14 | 3eqtr3a 2227 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1348 cid 4273 ccnv 4610 ccom 4615 wrel 4616 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 |
This theorem is referenced by: relcoi2 5141 funi 5230 fcoi2 5379 |
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