Theorem 2fcoidinvd 7250

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 7249	. 2 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
6	5, 2, 4	fcof1oinvd 7248	1 ⊢ (𝜑 → ◡𝐹 = 𝐺)

Syntax hints:   → wi 4   = wceq 1542   I cid 5525  ^◡ccnv 5630   ↾ cres 5633   ∘ ccom 5635  ⟶wf 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506

This theorem is referenced by:  fcof1o  7251  2fvidinvd  7254  pmtrfcnv  19439  qtophmeo  23782  fsovcnvd  44441