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Theorem f1oeq123d 5630
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 5624 . . 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 5625 . . 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 5626 . . 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 5412
This theorem is referenced by:  f1oprswap  5676  f1oprg  5677  cnfcom  7613  ackbij2lem2  8076  s2f1o  11818  s4f1o  11820  idffth  14085  ressffth  14090  indf1ofs  24376
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420
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