Theorem fcof1oinvd 6872

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 6875. (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 5579	. 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = (◡𝐹 ∘ ( I ↾ 𝐵)))
3		coass 5954	. . 3 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (◡𝐹 ∘ (𝐹 ∘ 𝐺))
4		fcof1oinvd.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
5		f1ococnv1 6469	. . . . . 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 5578	. . . 4 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺))
8		fcof1oinvd.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
9		fcoi2 6379	. . . . 5 ⊢ (𝐺:𝐵⟶𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
10	8, 9	syl 17	. . . 4 ⊢ (𝜑 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
11	7, 10	eqtrd 2807	. . 3 ⊢ (𝜑 → ((◡𝐹 ∘ 𝐹) ∘ 𝐺) = 𝐺)
12	3, 11	syl5eqr 2821	. 2 ⊢ (𝜑 → (◡𝐹 ∘ (𝐹 ∘ 𝐺)) = 𝐺)
13		f1ocnv 6453	. . . . 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 6441	. . . 4 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵⟶𝐴)
16	14, 15	syl 17	. . 3 ⊢ (𝜑 → ◡𝐹:𝐵⟶𝐴)
17		fcoi1 6378	. . 3 ⊢ (◡𝐹:𝐵⟶𝐴 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹)
18	16, 17	syl 17	. 2 ⊢ (𝜑 → (◡𝐹 ∘ ( I ↾ 𝐵)) = ◡𝐹)
19	2, 12, 18	3eqtr3rd 2816	1 ⊢ (𝜑 → ◡𝐹 = 𝐺)

Syntax hints:   → wi 4   = wceq 1508   I cid 5307  ^◡ccnv 5402   ↾ cres 5405   ∘ ccom 5407  ⟶wf 6181  –1-1-onto→wf1o 6184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192

This theorem is referenced by:  2fcoidinvd  6874