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Theorem fcof1oinvd 6872
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 6875. (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 5579 . 2 (𝜑 → (𝐹 ∘ (𝐹𝐺)) = (𝐹 ∘ ( I ↾ 𝐵)))
3 coass 5954 . . 3 ((𝐹𝐹) ∘ 𝐺) = (𝐹 ∘ (𝐹𝐺))
4 fcof1oinvd.f . . . . . 6 (𝜑𝐹:𝐴1-1-onto𝐵)
5 f1ococnv1 6469 . . . . . 6 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
64, 5syl 17 . . . . 5 (𝜑 → (𝐹𝐹) = ( I ↾ 𝐴))
76coeq1d 5578 . . . 4 (𝜑 → ((𝐹𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺))
8 fcof1oinvd.g . . . . 5 (𝜑𝐺:𝐵𝐴)
9 fcoi2 6379 . . . . 5 (𝐺:𝐵𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
108, 9syl 17 . . . 4 (𝜑 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
117, 10eqtrd 2807 . . 3 (𝜑 → ((𝐹𝐹) ∘ 𝐺) = 𝐺)
123, 11syl5eqr 2821 . 2 (𝜑 → (𝐹 ∘ (𝐹𝐺)) = 𝐺)
13 f1ocnv 6453 . . . . 5 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
144, 13syl 17 . . . 4 (𝜑𝐹:𝐵1-1-onto𝐴)
15 f1of 6441 . . . 4 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵𝐴)
1614, 15syl 17 . . 3 (𝜑𝐹:𝐵𝐴)
17 fcoi1 6378 . . 3 (𝐹:𝐵𝐴 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
1816, 17syl 17 . 2 (𝜑 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
192, 12, 183eqtr3rd 2816 1 (𝜑𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1508   I cid 5307  ccnv 5402  cres 5405  ccom 5407  wf 6181  1-1-ontowf1o 6184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192
This theorem is referenced by:  2fcoidinvd  6874
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