Theorem fcof1oinvd 7290

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 7293. (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 5837	. 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = (◡𝐹 ∘ ( I ↾ 𝐵)))
3		coass 6257	. . 3 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (◡𝐹 ∘ (𝐹 ∘ 𝐺))
4		fcof1oinvd.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
5		f1ococnv1 6843	. . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))
6	4, 5	syl 18	. . . . 5 ⊢ (𝜑 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))
7	6	coeq1d 5836	. . . 4 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺))
8		fcof1oinvd.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
9		fcoi2 6746	. . . . 5 ⊢ (𝐺:𝐵⟶𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
10	8, 9	syl 18	. . . 4 ⊢ (𝜑 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
11	7, 10	eqtrd 2795	. . 3 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = 𝐺)
12	3, 11	eqtr3id 2809	. 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = 𝐺)
13		f1ocnv 6826	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
14		f1of 6813	. . 3 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴)
15		fcoi1 6745	. . 3 ⊢ (◡𝐹:𝐵⟶𝐴 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹)
16	4, 13, 14, 15	4syl 20	. 2 ⊢ (𝜑 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹)
17	2, 12, 16	3eqtr3rd 2804	1 ⊢ (𝜑 → ◡𝐹 = 𝐺)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   I cid 5542  ^◡ccnv 5647   ↾ cres 5650   ∘ ccom 5652  ⟶wf 6524  –1-1-onto→wf1o 6527

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535

This theorem is used by:  2fcoidinvd  7292