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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 7243. (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 5819 | . 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = (◡𝐹 ∘ ( I ↾ 𝐵))) |
3 | coass 6218 | . . 3 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (◡𝐹 ∘ (𝐹 ∘ 𝐺)) | |
4 | fcof1oinvd.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | |
5 | f1ococnv1 6814 | . . . . . 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 5818 | . . . 4 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺)) |
8 | fcof1oinvd.g | . . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) | |
9 | fcoi2 6718 | . . . . 5 ⊢ (𝐺:𝐵⟶𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺) |
11 | 7, 10 | eqtrd 2773 | . . 3 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = 𝐺) |
12 | 3, 11 | eqtr3id 2787 | . 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = 𝐺) |
13 | f1ocnv 6797 | . . . . 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 6785 | . . . 4 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝜑 → ◡𝐹:𝐵⟶𝐴) |
17 | fcoi1 6717 | . . 3 ⊢ (◡𝐹:𝐵⟶𝐴 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹) | |
18 | 16, 17 | syl 17 | . 2 ⊢ (𝜑 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹) |
19 | 2, 12, 18 | 3eqtr3rd 2782 | 1 ⊢ (𝜑 → ◡𝐹 = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 I cid 5531 ◡ccnv 5633 ↾ cres 5636 ∘ ccom 5638 ⟶wf 6493 –1-1-onto→wf1o 6496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 |
This theorem is referenced by: 2fcoidinvd 7242 |
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