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Mirrors > Home > ILE Home > Th. List > coi2 | GIF 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 | ⊢ (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvco 4783 | . . 3 ⊢ ◡(◡𝐴 ∘ I ) = (◡ I ∘ ◡◡𝐴) | |
2 | relcnv 4976 | . . . . 5 ⊢ Rel ◡𝐴 | |
3 | coi1 5113 | . . . . 5 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ I ) = ◡𝐴) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (◡𝐴 ∘ I ) = ◡𝐴 |
5 | 4 | cnveqi 4773 | . . 3 ⊢ ◡(◡𝐴 ∘ I ) = ◡◡𝐴 |
6 | 1, 5 | eqtr3i 2187 | . 2 ⊢ (◡ I ∘ ◡◡𝐴) = ◡◡𝐴 |
7 | dfrel2 5048 | . . 3 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
8 | cnvi 5002 | . . . 4 ⊢ ◡ I = I | |
9 | coeq2 4756 | . . . . 5 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = (◡ I ∘ 𝐴)) | |
10 | coeq1 4755 | . . . . 5 ⊢ (◡ I = I → (◡ I ∘ 𝐴) = ( I ∘ 𝐴)) | |
11 | 9, 10 | sylan9eq 2217 | . . . 4 ⊢ ((◡◡𝐴 = 𝐴 ∧ ◡ I = I ) → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴)) |
12 | 8, 11 | mpan2 422 | . . 3 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴)) |
13 | 7, 12 | sylbi 120 | . 2 ⊢ (Rel 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴)) |
14 | 7 | biimpi 119 | . 2 ⊢ (Rel 𝐴 → ◡◡𝐴 = 𝐴) |
15 | 6, 13, 14 | 3eqtr3a 2221 | 1 ⊢ (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 I cid 4260 ◡ccnv 4597 ∘ ccom 4602 Rel 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-cnv 4606 df-co 4607 |
This theorem is referenced by: relcoi2 5128 funi 5214 fcoi2 5363 |
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