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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 5502 | . 2 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
2 | cnvco 5230 | . . . 4 ⊢ ◡(◡𝐴 ∘ I ) = (◡ I ∘ ◡◡𝐴) | |
3 | relcnv 5422 | . . . . . 6 ⊢ Rel ◡𝐴 | |
4 | coi1 5568 | . . . . . 6 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ I ) = ◡𝐴) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ (◡𝐴 ∘ I ) = ◡𝐴 |
6 | 5 | cnveqi 5219 | . . . 4 ⊢ ◡(◡𝐴 ∘ I ) = ◡◡𝐴 |
7 | 2, 6 | eqtr3i 2634 | . . 3 ⊢ (◡ I ∘ ◡◡𝐴) = ◡◡𝐴 |
8 | cnvi 5456 | . . . 4 ⊢ ◡ I = I | |
9 | coeq2 5202 | . . . . 5 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = (◡ I ∘ 𝐴)) | |
10 | coeq1 5201 | . . . . 5 ⊢ (◡ I = I → (◡ I ∘ 𝐴) = ( I ∘ 𝐴)) | |
11 | 9, 10 | sylan9eq 2664 | . . . 4 ⊢ ((◡◡𝐴 = 𝐴 ∧ ◡ I = I ) → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴)) |
12 | 8, 11 | mpan2 703 | . . 3 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴)) |
13 | id 22 | . . 3 ⊢ (◡◡𝐴 = 𝐴 → ◡◡𝐴 = 𝐴) | |
14 | 7, 12, 13 | 3eqtr3a 2668 | . 2 ⊢ (◡◡𝐴 = 𝐴 → ( I ∘ 𝐴) = 𝐴) |
15 | 1, 14 | sylbi 206 | 1 ⊢ (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 I cid 4948 ◡ccnv 5037 ∘ ccom 5042 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 |
This theorem is referenced by: relcoi2 5580 funi 5834 fcoi2 5992 |
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