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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 6185 | . 2 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
2 | cnvco 5883 | . . . 4 ⊢ ◡(◡𝐴 ∘ I ) = (◡ I ∘ ◡◡𝐴) | |
3 | relcnv 6100 | . . . . . 6 ⊢ Rel ◡𝐴 | |
4 | coi1 6258 | . . . . . 6 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ I ) = ◡𝐴) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ (◡𝐴 ∘ I ) = ◡𝐴 |
6 | 5 | cnveqi 5872 | . . . 4 ⊢ ◡(◡𝐴 ∘ I ) = ◡◡𝐴 |
7 | 2, 6 | eqtr3i 2762 | . . 3 ⊢ (◡ I ∘ ◡◡𝐴) = ◡◡𝐴 |
8 | cnvi 6138 | . . . 4 ⊢ ◡ I = I | |
9 | coeq2 5856 | . . . . 5 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = (◡ I ∘ 𝐴)) | |
10 | coeq1 5855 | . . . . 5 ⊢ (◡ I = I → (◡ I ∘ 𝐴) = ( I ∘ 𝐴)) | |
11 | 9, 10 | sylan9eq 2792 | . . . 4 ⊢ ((◡◡𝐴 = 𝐴 ∧ ◡ I = I ) → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴)) |
12 | 8, 11 | mpan2 689 | . . 3 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴)) |
13 | id 22 | . . 3 ⊢ (◡◡𝐴 = 𝐴 → ◡◡𝐴 = 𝐴) | |
14 | 7, 12, 13 | 3eqtr3a 2796 | . 2 ⊢ (◡◡𝐴 = 𝐴 → ( I ∘ 𝐴) = 𝐴) |
15 | 1, 14 | sylbi 216 | 1 ⊢ (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 I cid 5572 ◡ccnv 5674 ∘ ccom 5679 Rel wrel 5680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 |
This theorem is referenced by: relcoi2 6273 funi 6577 fcoi2 6763 |
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