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Mirrors > Home > MPE Home > Th. List > coi2 | Structured version Visualization version 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 | dfrel2 6211 | . 2 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
2 | cnvco 5899 | . . . 4 ⊢ ◡(◡𝐴 ∘ I ) = (◡ I ∘ ◡◡𝐴) | |
3 | relcnv 6125 | . . . . . 6 ⊢ Rel ◡𝐴 | |
4 | coi1 6284 | . . . . . 6 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ I ) = ◡𝐴) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ (◡𝐴 ∘ I ) = ◡𝐴 |
6 | 5 | cnveqi 5888 | . . . 4 ⊢ ◡(◡𝐴 ∘ I ) = ◡◡𝐴 |
7 | 2, 6 | eqtr3i 2765 | . . 3 ⊢ (◡ I ∘ ◡◡𝐴) = ◡◡𝐴 |
8 | cnvi 6164 | . . . 4 ⊢ ◡ I = I | |
9 | coeq2 5872 | . . . . 5 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = (◡ I ∘ 𝐴)) | |
10 | coeq1 5871 | . . . . 5 ⊢ (◡ I = I → (◡ I ∘ 𝐴) = ( I ∘ 𝐴)) | |
11 | 9, 10 | sylan9eq 2795 | . . . 4 ⊢ ((◡◡𝐴 = 𝐴 ∧ ◡ I = I ) → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴)) |
12 | 8, 11 | mpan2 691 | . . 3 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴)) |
13 | id 22 | . . 3 ⊢ (◡◡𝐴 = 𝐴 → ◡◡𝐴 = 𝐴) | |
14 | 7, 12, 13 | 3eqtr3a 2799 | . 2 ⊢ (◡◡𝐴 = 𝐴 → ( I ∘ 𝐴) = 𝐴) |
15 | 1, 14 | sylbi 217 | 1 ⊢ (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 I cid 5582 ◡ccnv 5688 ∘ ccom 5693 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 |
This theorem is referenced by: relcoi2 6299 funi 6600 fcoi2 6784 |
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