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| 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 6208 | . 2 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
| 2 | cnvco 5895 | . . . 4 ⊢ ◡(◡𝐴 ∘ I ) = (◡ I ∘ ◡◡𝐴) | |
| 3 | relcnv 6121 | . . . . . 6 ⊢ Rel ◡𝐴 | |
| 4 | coi1 6281 | . . . . . 6 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ I ) = ◡𝐴) | |
| 5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ (◡𝐴 ∘ I ) = ◡𝐴 | 
| 6 | 5 | cnveqi 5884 | . . . 4 ⊢ ◡(◡𝐴 ∘ I ) = ◡◡𝐴 | 
| 7 | 2, 6 | eqtr3i 2766 | . . 3 ⊢ (◡ I ∘ ◡◡𝐴) = ◡◡𝐴 | 
| 8 | cnvi 6160 | . . . 4 ⊢ ◡ I = I | |
| 9 | coeq2 5868 | . . . . 5 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = (◡ I ∘ 𝐴)) | |
| 10 | coeq1 5867 | . . . . 5 ⊢ (◡ I = I → (◡ I ∘ 𝐴) = ( I ∘ 𝐴)) | |
| 11 | 9, 10 | sylan9eq 2796 | . . . 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 2800 | . 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 1539 I cid 5576 ◡ccnv 5683 ∘ ccom 5688 Rel wrel 5689 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 | 
| This theorem is referenced by: relcoi2 6296 funi 6597 fcoi2 6782 | 
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