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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 6875. (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 5579 | . 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = (◡𝐹 ∘ ( I ↾ 𝐵))) |
3 | coass 5954 | . . 3 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (◡𝐹 ∘ (𝐹 ∘ 𝐺)) | |
4 | fcof1oinvd.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | |
5 | f1ococnv1 6469 | . . . . . 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 5578 | . . . 4 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺)) |
8 | fcof1oinvd.g | . . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) | |
9 | fcoi2 6379 | . . . . 5 ⊢ (𝐺:𝐵⟶𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺) |
11 | 7, 10 | eqtrd 2807 | . . 3 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = 𝐺) |
12 | 3, 11 | syl5eqr 2821 | . 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = 𝐺) |
13 | f1ocnv 6453 | . . . . 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 6441 | . . . 4 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝜑 → ◡𝐹:𝐵⟶𝐴) |
17 | fcoi1 6378 | . . 3 ⊢ (◡𝐹:𝐵⟶𝐴 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹) | |
18 | 16, 17 | syl 17 | . 2 ⊢ (𝜑 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹) |
19 | 2, 12, 18 | 3eqtr3rd 2816 | 1 ⊢ (𝜑 → ◡𝐹 = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 I cid 5307 ◡ccnv 5402 ↾ cres 5405 ∘ ccom 5407 ⟶wf 6181 –1-1-onto→wf1o 6184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4926 df-opab 4988 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 |
This theorem is referenced by: 2fcoidinvd 6874 |
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