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Theorem f1oeq123d 5663
Description: Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
f1eq123d.1  |-  ( ph  ->  F  =  G )
f1eq123d.2  |-  ( ph  ->  A  =  B )
f1eq123d.3  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
f1oeq123d  |-  ( ph  ->  ( F : A -1-1-onto-> C  <->  G : B -1-1-onto-> D ) )

Proof of Theorem f1oeq123d
StepHypRef Expression
1 f1eq123d.1 . . 3  |-  ( ph  ->  F  =  G )
2 f1oeq1 5657 . . 3  |-  ( F  =  G  ->  ( F : A -1-1-onto-> C  <->  G : A -1-1-onto-> C ) )
31, 2syl 16 . 2  |-  ( ph  ->  ( F : A -1-1-onto-> C  <->  G : A -1-1-onto-> C ) )
4 f1eq123d.2 . . 3  |-  ( ph  ->  A  =  B )
5 f1oeq2 5658 . . 3  |-  ( A  =  B  ->  ( G : A -1-1-onto-> C  <->  G : B -1-1-onto-> C ) )
64, 5syl 16 . 2  |-  ( ph  ->  ( G : A -1-1-onto-> C  <->  G : B -1-1-onto-> C ) )
7 f1eq123d.3 . . 3  |-  ( ph  ->  C  =  D )
8 f1oeq3 5659 . . 3  |-  ( C  =  D  ->  ( G : B -1-1-onto-> C  <->  G : B -1-1-onto-> D ) )
97, 8syl 16 . 2  |-  ( ph  ->  ( G : B -1-1-onto-> C  <->  G : B -1-1-onto-> D ) )
103, 6, 93bitrd 271 1  |-  ( ph  ->  ( F : A -1-1-onto-> C  <->  G : B -1-1-onto-> D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652   -1-1-onto->wf1o 5445
This theorem is referenced by:  f1oprswap  5709  f1oprg  5710  cnfcom  7649  ackbij2lem2  8112  s2f1o  11855  s4f1o  11857  idffth  14122  ressffth  14127  indf1ofs  24415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453
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