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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 7043. (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 5726 | . 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = (◡𝐹 ∘ ( I ↾ 𝐵))) |
3 | coass 6111 | . . 3 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (◡𝐹 ∘ (𝐹 ∘ 𝐺)) | |
4 | fcof1oinvd.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | |
5 | f1ococnv1 6636 | . . . . . 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 5725 | . . . 4 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺)) |
8 | fcof1oinvd.g | . . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) | |
9 | fcoi2 6546 | . . . . 5 ⊢ (𝐺:𝐵⟶𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺) |
11 | 7, 10 | eqtrd 2853 | . . 3 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = 𝐺) |
12 | 3, 11 | syl5eqr 2867 | . 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = 𝐺) |
13 | f1ocnv 6620 | . . . . 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 6608 | . . . 4 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝜑 → ◡𝐹:𝐵⟶𝐴) |
17 | fcoi1 6545 | . . 3 ⊢ (◡𝐹:𝐵⟶𝐴 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹) | |
18 | 16, 17 | syl 17 | . 2 ⊢ (𝜑 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹) |
19 | 2, 12, 18 | 3eqtr3rd 2862 | 1 ⊢ (𝜑 → ◡𝐹 = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 I cid 5452 ◡ccnv 5547 ↾ cres 5550 ∘ ccom 5552 ⟶wf 6344 –1-1-onto→wf1o 6347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 |
This theorem is referenced by: 2fcoidinvd 7042 |
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