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Theorem f1oeq2 5325
Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
Assertion
Ref Expression
f1oeq2  |-  ( A  =  B  ->  ( F : A -1-1-onto-> C  <->  F : B -1-1-onto-> C ) )

Proof of Theorem f1oeq2
StepHypRef Expression
1 f1eq2 5292 . . 3  |-  ( A  =  B  ->  ( F : A -1-1-> C  <->  F : B -1-1-> C ) )
2 foeq2 5310 . . 3  |-  ( A  =  B  ->  ( F : A -onto-> C  <->  F : B -onto-> C ) )
31, 2anbi12d 462 . 2  |-  ( A  =  B  ->  (
( F : A -1-1-> C  /\  F : A -onto-> C )  <->  ( F : B -1-1-> C  /\  F : B -onto-> C ) ) )
4 df-f1o 5098 . 2  |-  ( F : A -1-1-onto-> C  <->  ( F : A -1-1-> C  /\  F : A -onto-> C ) )
5 df-f1o 5098 . 2  |-  ( F : B -1-1-onto-> C  <->  ( F : B -1-1-> C  /\  F : B -onto-> C ) )
63, 4, 53bitr4g 222 1  |-  ( A  =  B  ->  ( F : A -1-1-onto-> C  <->  F : B -1-1-onto-> C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314   -1-1->wf1 5088   -onto->wfo 5089   -1-1-onto->wf1o 5090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-4 1470  ax-17 1489  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-cleq 2108  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098
This theorem is referenced by:  f1oeq23  5327  f1oeq123d  5330  f1osng  5374  isoeq4  5671  bren  6607  f1dmvrnfibi  6798  summodclem3  11089  summodclem2a  11090  summodc  11092  fsum3  11096  fsumf1o  11099  sumsnf  11118
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