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Theorem f1oeq23 5354
Description: Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)
Assertion
Ref Expression
f1oeq23  |-  ( ( A  =  B  /\  C  =  D )  ->  ( F : A -1-1-onto-> C  <->  F : B -1-1-onto-> D ) )

Proof of Theorem f1oeq23
StepHypRef Expression
1 f1oeq2 5352 . 2  |-  ( A  =  B  ->  ( F : A -1-1-onto-> C  <->  F : B -1-1-onto-> C ) )
2 f1oeq3 5353 . 2  |-  ( C  =  D  ->  ( F : B -1-1-onto-> C  <->  F : B -1-1-onto-> D ) )
31, 2sylan9bb 457 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( F : A -1-1-onto-> C  <->  F : B -1-1-onto-> D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331   -1-1-onto->wf1o 5117
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-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-in 3072  df-ss 3079  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125
This theorem is referenced by:  zfz1isolem1  10576
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