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Theorem fcof1oinvd 7040
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 7043. (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 5726 . 2 (𝜑 → (𝐹 ∘ (𝐹𝐺)) = (𝐹 ∘ ( I ↾ 𝐵)))
3 coass 6111 . . 3 ((𝐹𝐹) ∘ 𝐺) = (𝐹 ∘ (𝐹𝐺))
4 fcof1oinvd.f . . . . . 6 (𝜑𝐹:𝐴1-1-onto𝐵)
5 f1ococnv1 6636 . . . . . 6 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
64, 5syl 17 . . . . 5 (𝜑 → (𝐹𝐹) = ( I ↾ 𝐴))
76coeq1d 5725 . . . 4 (𝜑 → ((𝐹𝐹) ∘ 𝐺) = (( I ↾ 𝐴) ∘ 𝐺))
8 fcof1oinvd.g . . . . 5 (𝜑𝐺:𝐵𝐴)
9 fcoi2 6546 . . . . 5 (𝐺:𝐵𝐴 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
108, 9syl 17 . . . 4 (𝜑 → (( I ↾ 𝐴) ∘ 𝐺) = 𝐺)
117, 10eqtrd 2853 . . 3 (𝜑 → ((𝐹𝐹) ∘ 𝐺) = 𝐺)
123, 11syl5eqr 2867 . 2 (𝜑 → (𝐹 ∘ (𝐹𝐺)) = 𝐺)
13 f1ocnv 6620 . . . . 5 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
144, 13syl 17 . . . 4 (𝜑𝐹:𝐵1-1-onto𝐴)
15 f1of 6608 . . . 4 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵𝐴)
1614, 15syl 17 . . 3 (𝜑𝐹:𝐵𝐴)
17 fcoi1 6545 . . 3 (𝐹:𝐵𝐴 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
1816, 17syl 17 . 2 (𝜑 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
192, 12, 183eqtr3rd 2862 1 (𝜑𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528   I cid 5452  ccnv 5547  cres 5550  ccom 5552  wf 6344  1-1-ontowf1o 6347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355
This theorem is referenced by:  2fcoidinvd  7042
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