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Theorem f1oeq2d 5363
Description: Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
f1oeq2d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
f1oeq2d  |-  ( ph  ->  ( F : A -1-1-onto-> C  <->  F : B -1-1-onto-> C ) )

Proof of Theorem f1oeq2d
StepHypRef Expression
1 f1oeq2d.1 . 2  |-  ( ph  ->  A  =  B )
2 f1oeq2 5357 . 2  |-  ( A  =  B  ->  ( F : A -1-1-onto-> C  <->  F : B -1-1-onto-> C ) )
31, 2syl 14 1  |-  ( ph  ->  ( F : A -1-1-onto-> C  <->  F : B -1-1-onto-> C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331   -1-1-onto->wf1o 5122
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 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130
This theorem is referenced by:  prodmodclem3  11356  prodmodc  11359
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