Theorem 2fcoidinvd 7289

Description: Show that a function is the inverse of a function if their compositions are the identity functions. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.)

Hypotheses

Ref	Expression
fcof1od.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fcof1od.g	⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
fcof1od.a	⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))
fcof1od.b	⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵))

Assertion

Ref	Expression
2fcoidinvd	⊢ (𝜑 → ◡𝐹 = 𝐺)

Proof of Theorem 2fcoidinvd

Step	Hyp	Ref	Expression
1		fcof1od.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		fcof1od.g	. . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
3		fcof1od.a	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))
4		fcof1od.b	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵))
5	1, 2, 3, 4	fcof1od 7288	. 2 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
6	5, 2, 4	fcof1oinvd 7287	1 ⊢ (𝜑 → ◡𝐹 = 𝐺)

Syntax hints:   → wi 4   = wceq 1533   I cid 5566  ^◡ccnv 5668   ↾ cres 5671   ∘ ccom 5673  ⟶wf 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is referenced by:  fcof1o  7290  2fvidinvd  7293  pmtrfcnv  19384  qtophmeo  23676  fsovcnvd  43341