MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1oeq23 Unicode version

Theorem f1oeq23 5466
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 5464 . 2  |-  ( A  =  B  ->  ( F : A -1-1-onto-> C  <->  F : B -1-1-onto-> C ) )
2 f1oeq3 5465 . 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 1623   -1-1-onto->wf1o 5254
This theorem is referenced by:  ackbij2lem2  7866  seqf1o  11087  eulerthlem2  12850  isgim  14726  symgval  14771  islmim  15815  isoriso  25212  phckle  26027  psckle  26028  eldioph2lem1  26839  enfixsn  27257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262
  Copyright terms: Public domain W3C validator