Theorem fcof1oinvd 7103

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 7106. (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 5731	. 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = (◡𝐹 ∘ ( I ↾ 𝐵)))
3		coass 6129	. . 3 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (◡𝐹 ∘ (𝐹 ∘ 𝐺))
4		fcof1oinvd.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
5		f1ococnv1 6689	. . . . . 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 5730	. . . 4 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺))
8		fcof1oinvd.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
9		fcoi2 6594	. . . . 5 ⊢ (𝐺:𝐵⟶𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
11	7, 10	eqtrd 2777	. . 3 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = 𝐺)
12	3, 11	eqtr3id 2792	. 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = 𝐺)
13		f1ocnv 6673	. . . . 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 6661	. . . 4 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴)
16	14, 15	syl 17	. . 3 ⊢ (𝜑 → ◡𝐹:𝐵⟶𝐴)
17		fcoi1 6593	. . 3 ⊢ (◡𝐹:𝐵⟶𝐴 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹)
18	16, 17	syl 17	. 2 ⊢ (𝜑 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹)
19	2, 12, 18	3eqtr3rd 2786	1 ⊢ (𝜑 → ◡𝐹 = 𝐺)

Syntax hints:   → wi 4   = wceq 1543   I cid 5454  ^◡ccnv 5550   ↾ cres 5553   ∘ ccom 5555  ⟶wf 6376  –1-1-onto→wf1o 6379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387

This theorem is referenced by:  2fcoidinvd  7105