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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 6040 | . 2 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴) | |
2 | cnvco 5750 | . . . 4 ⊢ ◡(◡𝐴 ∘ I ) = (◡ I ∘ ◡◡𝐴) | |
3 | relcnv 5961 | . . . . . 6 ⊢ Rel ◡𝐴 | |
4 | coi1 6109 | . . . . . 6 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ I ) = ◡𝐴) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ (◡𝐴 ∘ I ) = ◡𝐴 |
6 | 5 | cnveqi 5739 | . . . 4 ⊢ ◡(◡𝐴 ∘ I ) = ◡◡𝐴 |
7 | 2, 6 | eqtr3i 2846 | . . 3 ⊢ (◡ I ∘ ◡◡𝐴) = ◡◡𝐴 |
8 | cnvi 5994 | . . . 4 ⊢ ◡ I = I | |
9 | coeq2 5723 | . . . . 5 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = (◡ I ∘ 𝐴)) | |
10 | coeq1 5722 | . . . . 5 ⊢ (◡ I = I → (◡ I ∘ 𝐴) = ( I ∘ 𝐴)) | |
11 | 9, 10 | sylan9eq 2876 | . . . 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 2880 | . 2 ⊢ (◡◡𝐴 = 𝐴 → ( I ∘ 𝐴) = 𝐴) |
15 | 1, 14 | sylbi 219 | 1 ⊢ (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 I cid 5453 ◡ccnv 5548 ∘ ccom 5553 Rel wrel 5554 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 |
This theorem is referenced by: relcoi2 6122 funi 6381 fcoi2 6547 |
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