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Theorem f1oeq23 5572
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 5570 . 2  |-  ( A  =  B  ->  ( F : A -1-1-onto-> C  <->  F : B -1-1-onto-> C ) )
2 f1oeq3 5571 . 2  |-  ( C  =  D  ->  ( F : B -1-1-onto-> C  <->  F : B -1-1-onto-> D ) )
31, 2sylan9bb 680 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    <-> wb 176    /\ wa 358    = wceq 1647   -1-1-onto->wf1o 5357
This theorem is referenced by:  ackbij2lem2  8013  seqf1o  11251  eulerthlem2  13058  isgim  14936  symgval  14981  islmim  16025  eldioph2lem1  26345  enfixsn  26763
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-in 3245  df-ss 3252  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365
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