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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 6451. (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 5206 | . 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = (◡𝐹 ∘ ( I ↾ 𝐵))) |
3 | coass 5571 | . . 3 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (◡𝐹 ∘ (𝐹 ∘ 𝐺)) | |
4 | fcof1oinvd.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | |
5 | f1ococnv1 6078 | . . . . . 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 5205 | . . . 4 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺)) |
8 | fcof1oinvd.g | . . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) | |
9 | fcoi2 5992 | . . . . 5 ⊢ (𝐺:𝐵⟶𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺) |
11 | 7, 10 | eqtrd 2644 | . . 3 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = 𝐺) |
12 | 3, 11 | syl5eqr 2658 | . 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = 𝐺) |
13 | f1ocnv 6062 | . . . . 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 6050 | . . . 4 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝜑 → ◡𝐹:𝐵⟶𝐴) |
17 | fcoi1 5991 | . . 3 ⊢ (◡𝐹:𝐵⟶𝐴 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹) | |
18 | 16, 17 | syl 17 | . 2 ⊢ (𝜑 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹) |
19 | 2, 12, 18 | 3eqtr3rd 2653 | 1 ⊢ (𝜑 → ◡𝐹 = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 I cid 4948 ◡ccnv 5037 ↾ cres 5040 ∘ ccom 5042 ⟶wf 5800 –1-1-onto→wf1o 5803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 |
This theorem is referenced by: 2fcoidinvd 6450 |
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