Theorem 2fcoidinvd 7300

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 7299	. 2 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
6	5, 2, 4	fcof1oinvd 7298	1 ⊢ (𝜑 → ◡𝐹 = 𝐺)

Syntax hints:   → wi 4   = wceq 1533   I cid 5569  ^◡ccnv 5671   ↾ cres 5674   ∘ ccom 5676  ⟶wf 6539

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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551

This theorem is referenced by:  fcof1o  7301  2fvidinvd  7304  pmtrfcnv  19423  qtophmeo  23739  fsovcnvd  43509