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Theorem f1oeq123d 5604
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 5598 . . 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 5599 . . 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 5600 . . 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 1649   -1-1-onto->wf1o 5386
This theorem is referenced by:  f1oprswap  5650  f1oprg  5651  cnfcom  7583  ackbij2lem2  8046  s2f1o  11783  s4f1o  11785  idffth  14050  ressffth  14055  indf1ofs  24212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394
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