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Theorem f1oeq23 5744
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 5742 . 2  |-  ( A  =  B  ->  ( F : A -1-1-onto-> C  <->  F : B -1-1-onto-> C ) )
2 f1oeq3 5743 . 2  |-  ( C  =  D  ->  ( F : B -1-1-onto-> C  <->  F : B -1-1-onto-> D ) )
31, 2sylan9bb 699 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( F : A -1-1-onto-> C  <->  F : B -1-1-onto-> D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   -1-1-onto->wf1o 5526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-in 3444  df-ss 3451  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534
This theorem is referenced by:  enfixsn  7531  ackbij2lem2  8521  seqf1o  11965  eulerthlem2  13976  isgim  15910  symgval  16004  islmim  17267  fpwrelmapffs  26186  eldioph2lem1  29247
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