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Theorem 2fcoidinvd 6547
Description: Show that a function is the inverse of a function if their compositions are the identity functions. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.)
Hypotheses
Ref Expression
fcof1od.f (𝜑𝐹:𝐴𝐵)
fcof1od.g (𝜑𝐺:𝐵𝐴)
fcof1od.a (𝜑 → (𝐺𝐹) = ( I ↾ 𝐴))
fcof1od.b (𝜑 → (𝐹𝐺) = ( I ↾ 𝐵))
Assertion
Ref Expression
2fcoidinvd (𝜑𝐹 = 𝐺)

Proof of Theorem 2fcoidinvd
StepHypRef Expression
1 fcof1od.f . . 3 (𝜑𝐹:𝐴𝐵)
2 fcof1od.g . . 3 (𝜑𝐺:𝐵𝐴)
3 fcof1od.a . . 3 (𝜑 → (𝐺𝐹) = ( I ↾ 𝐴))
4 fcof1od.b . . 3 (𝜑 → (𝐹𝐺) = ( I ↾ 𝐵))
51, 2, 3, 4fcof1od 6546 . 2 (𝜑𝐹:𝐴1-1-onto𝐵)
65, 2, 4fcof1oinvd 6545 1 (𝜑𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482   I cid 5021  ccnv 5111  cres 5114  ccom 5116  wf 5882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894
This theorem is referenced by:  fcof1o  6548  2fvidinvd  6551  pmtrfcnv  17878  qtophmeo  21614  fsovcnvd  38134
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