Theorem fcof1oinvd 7268

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 7271. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.)

Hypotheses

Ref	Expression
fcof1oinvd.f	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
fcof1oinvd.g	⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
fcof1oinvd.b	⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵))

Assertion

Ref	Expression
fcof1oinvd	⊢ (𝜑 → ◡𝐹 = 𝐺)

Proof of Theorem fcof1oinvd

Step	Hyp	Ref	Expression
1		fcof1oinvd.b	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵))
2	1	coeq2d 5826	. 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = (◡𝐹 ∘ ( I ↾ 𝐵)))
3		coass 6238	. . 3 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (◡𝐹 ∘ (𝐹 ∘ 𝐺))
4		fcof1oinvd.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
5		f1ococnv1 6829	. . . . . 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 5825	. . . 4 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺))
8		fcof1oinvd.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
9		fcoi2 6735	. . . . 5 ⊢ (𝐺:𝐵⟶𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
11	7, 10	eqtrd 2764	. . 3 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = 𝐺)
12	3, 11	eqtr3id 2778	. 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = 𝐺)
13		f1ocnv 6812	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
14		f1of 6800	. . 3 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴)
15		fcoi1 6734	. . 3 ⊢ (◡𝐹:𝐵⟶𝐴 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹)
16	4, 13, 14, 15	4syl 19	. 2 ⊢ (𝜑 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹)
17	2, 12, 16	3eqtr3rd 2773	1 ⊢ (𝜑 → ◡𝐹 = 𝐺)

Syntax hints:   → wi 4   = wceq 1540   I cid 5532  ^◡ccnv 5637   ↾ cres 5640   ∘ ccom 5642  ⟶wf 6507  –1-1-onto→wf1o 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518

This theorem is referenced by:  2fcoidinvd  7270