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Theorem 2fcoidinvd 7300
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 7299 . 2 (𝜑𝐹:𝐴1-1-onto𝐵)
65, 2, 4fcof1oinvd 7298 1 (𝜑𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533   I cid 5569  ccnv 5671  cres 5674  ccom 5676  wf 6539
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551
This theorem is referenced by:  fcof1o  7301  2fvidinvd  7304  pmtrfcnv  19423  qtophmeo  23739  fsovcnvd  43509
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