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Theorem f1oeq2 5239
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 5206 . . 3  |-  ( A  =  B  ->  ( F : A -1-1-> C  <->  F : B -1-1-> C ) )
2 foeq2 5224 . . 3  |-  ( A  =  B  ->  ( F : A -onto-> C  <->  F : B -onto-> C ) )
31, 2anbi12d 457 . 2  |-  ( A  =  B  ->  (
( F : A -1-1-> C  /\  F : A -onto-> C )  <->  ( F : B -1-1-> C  /\  F : B -onto-> C ) ) )
4 df-f1o 5017 . 2  |-  ( F : A -1-1-onto-> C  <->  ( F : A -1-1-> C  /\  F : A -onto-> C ) )
5 df-f1o 5017 . 2  |-  ( F : B -1-1-onto-> C  <->  ( F : B -1-1-> C  /\  F : B -onto-> C ) )
63, 4, 53bitr4g 221 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 102    <-> wb 103    = wceq 1289   -1-1->wf1 5007   -onto->wfo 5008   -1-1-onto->wf1o 5009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017
This theorem is referenced by:  f1oeq23  5241  f1oeq123d  5244  f1osng  5288  isoeq4  5575  bren  6454  f1dmvrnfibi  6643  isummolem3  10757  isummolem2a  10758  isummo  10760  fisum  10765  fsumf1o  10769  sumsnf  10790
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