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Theorem fcof1oinvd 7287
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 7290. (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 5856 . 2 (𝜑 → (𝐹 ∘ (𝐹𝐺)) = (𝐹 ∘ ( I ↾ 𝐵)))
3 coass 6258 . . 3 ((𝐹𝐹) ∘ 𝐺) = (𝐹 ∘ (𝐹𝐺))
4 fcof1oinvd.f . . . . . 6 (𝜑𝐹:𝐴1-1-onto𝐵)
5 f1ococnv1 6856 . . . . . 6 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
64, 5syl 17 . . . . 5 (𝜑 → (𝐹𝐹) = ( I ↾ 𝐴))
76coeq1d 5855 . . . 4 (𝜑 → ((𝐹𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺))
8 fcof1oinvd.g . . . . 5 (𝜑𝐺:𝐵𝐴)
9 fcoi2 6760 . . . . 5 (𝐺:𝐵𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
108, 9syl 17 . . . 4 (𝜑 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
117, 10eqtrd 2766 . . 3 (𝜑 → ((𝐹𝐹) ∘ 𝐺) = 𝐺)
123, 11eqtr3id 2780 . 2 (𝜑 → (𝐹 ∘ (𝐹𝐺)) = 𝐺)
13 f1ocnv 6839 . . . . 5 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
144, 13syl 17 . . . 4 (𝜑𝐹:𝐵1-1-onto𝐴)
15 f1of 6827 . . . 4 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵𝐴)
1614, 15syl 17 . . 3 (𝜑𝐹:𝐵𝐴)
17 fcoi1 6759 . . 3 (𝐹:𝐵𝐴 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
1816, 17syl 17 . 2 (𝜑 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
192, 12, 183eqtr3rd 2775 1 (𝜑𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533   I cid 5566  ccnv 5668  cres 5671  ccom 5673  wf 6533  1-1-ontowf1o 6536
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-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-br 5142  df-opab 5204  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-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544
This theorem is referenced by:  2fcoidinvd  7289
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