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Theorem fcof1oinvd 6503
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 6506. (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 5249 . 2 (𝜑 → (𝐹 ∘ (𝐹𝐺)) = (𝐹 ∘ ( I ↾ 𝐵)))
3 coass 5616 . . 3 ((𝐹𝐹) ∘ 𝐺) = (𝐹 ∘ (𝐹𝐺))
4 fcof1oinvd.f . . . . . 6 (𝜑𝐹:𝐴1-1-onto𝐵)
5 f1ococnv1 6124 . . . . . 6 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
64, 5syl 17 . . . . 5 (𝜑 → (𝐹𝐹) = ( I ↾ 𝐴))
76coeq1d 5248 . . . 4 (𝜑 → ((𝐹𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺))
8 fcof1oinvd.g . . . . 5 (𝜑𝐺:𝐵𝐴)
9 fcoi2 6038 . . . . 5 (𝐺:𝐵𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
108, 9syl 17 . . . 4 (𝜑 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
117, 10eqtrd 2660 . . 3 (𝜑 → ((𝐹𝐹) ∘ 𝐺) = 𝐺)
123, 11syl5eqr 2674 . 2 (𝜑 → (𝐹 ∘ (𝐹𝐺)) = 𝐺)
13 f1ocnv 6108 . . . . 5 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
144, 13syl 17 . . . 4 (𝜑𝐹:𝐵1-1-onto𝐴)
15 f1of 6096 . . . 4 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵𝐴)
1614, 15syl 17 . . 3 (𝜑𝐹:𝐵𝐴)
17 fcoi1 6037 . . 3 (𝐹:𝐵𝐴 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
1816, 17syl 17 . 2 (𝜑 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
192, 12, 183eqtr3rd 2669 1 (𝜑𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480   I cid 4989  ccnv 5078  cres 5081  ccom 5083  wf 5846  1-1-ontowf1o 5849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857
This theorem is referenced by:  2fcoidinvd  6505
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