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Theorem fcof1oinvd 7314
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 7317. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.)
Hypotheses
Ref Expression
fcof1oinvd.f (𝜑𝐹:𝐴1-1-onto𝐵)
fcof1oinvd.g (𝜑𝐺:𝐵𝐴)
fcof1oinvd.b (𝜑 → (𝐹𝐺) = ( I ↾ 𝐵))
Assertion
Ref Expression
fcof1oinvd (𝜑𝐹 = 𝐺)

Proof of Theorem fcof1oinvd
StepHypRef Expression
1 fcof1oinvd.b . . 3 (𝜑 → (𝐹𝐺) = ( I ↾ 𝐵))
21coeq2d 5872 . 2 (𝜑 → (𝐹 ∘ (𝐹𝐺)) = (𝐹 ∘ ( I ↾ 𝐵)))
3 coass 6284 . . 3 ((𝐹𝐹) ∘ 𝐺) = (𝐹 ∘ (𝐹𝐺))
4 fcof1oinvd.f . . . . . 6 (𝜑𝐹:𝐴1-1-onto𝐵)
5 f1ococnv1 6876 . . . . . 6 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
64, 5syl 17 . . . . 5 (𝜑 → (𝐹𝐹) = ( I ↾ 𝐴))
76coeq1d 5871 . . . 4 (𝜑 → ((𝐹𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺))
8 fcof1oinvd.g . . . . 5 (𝜑𝐺:𝐵𝐴)
9 fcoi2 6782 . . . . 5 (𝐺:𝐵𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
108, 9syl 17 . . . 4 (𝜑 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
117, 10eqtrd 2776 . . 3 (𝜑 → ((𝐹𝐹) ∘ 𝐺) = 𝐺)
123, 11eqtr3id 2790 . 2 (𝜑 → (𝐹 ∘ (𝐹𝐺)) = 𝐺)
13 f1ocnv 6859 . . 3 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
14 f1of 6847 . . 3 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵𝐴)
15 fcoi1 6781 . . 3 (𝐹:𝐵𝐴 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
164, 13, 14, 154syl 19 . 2 (𝜑 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
172, 12, 163eqtr3rd 2785 1 (𝜑𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539   I cid 5576  ccnv 5683  cres 5686  ccom 5688  wf 6556  1-1-ontowf1o 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567
This theorem is referenced by:  2fcoidinvd  7316
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