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Mirrors > Home > MPE Home > Th. List > fcof1oinvd | Structured version Visualization version GIF version |
Description: Show that a function is the inverse of a bijective function if their composition is the identity function. Formerly part of proof of fcof1o 7290. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.) |
Ref | Expression |
---|---|
fcof1oinvd.f | ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
fcof1oinvd.g | ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) |
fcof1oinvd.b | ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵)) |
Ref | Expression |
---|---|
fcof1oinvd | ⊢ (𝜑 → ◡𝐹 = 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcof1oinvd.b | . . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵)) | |
2 | 1 | coeq2d 5856 | . 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = (◡𝐹 ∘ ( I ↾ 𝐵))) |
3 | coass 6258 | . . 3 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (◡𝐹 ∘ (𝐹 ∘ 𝐺)) | |
4 | fcof1oinvd.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | |
5 | f1ococnv1 6856 | . . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) |
7 | 6 | coeq1d 5855 | . . . 4 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺)) |
8 | fcof1oinvd.g | . . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) | |
9 | fcoi2 6760 | . . . . 5 ⊢ (𝐺:𝐵⟶𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺) |
11 | 7, 10 | eqtrd 2766 | . . 3 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = 𝐺) |
12 | 3, 11 | eqtr3id 2780 | . 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = 𝐺) |
13 | f1ocnv 6839 | . . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) | |
14 | 4, 13 | syl 17 | . . . 4 ⊢ (𝜑 → ◡𝐹:𝐵–1-1-onto→𝐴) |
15 | f1of 6827 | . . . 4 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝜑 → ◡𝐹:𝐵⟶𝐴) |
17 | fcoi1 6759 | . . 3 ⊢ (◡𝐹:𝐵⟶𝐴 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹) | |
18 | 16, 17 | syl 17 | . 2 ⊢ (𝜑 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹) |
19 | 2, 12, 18 | 3eqtr3rd 2775 | 1 ⊢ (𝜑 → ◡𝐹 = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 I cid 5566 ◡ccnv 5668 ↾ cres 5671 ∘ ccom 5673 ⟶wf 6533 –1-1-onto→wf1o 6536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 |
This theorem is referenced by: 2fcoidinvd 7289 |
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