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Theorem f1oeq123d 5549
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 5543 . . 3  |-  ( F  =  G  ->  ( F : A -1-1-onto-> C  <->  G : A -1-1-onto-> C ) )
31, 2syl 15 . 2  |-  ( ph  ->  ( F : A -1-1-onto-> C  <->  G : A -1-1-onto-> C ) )
4 f1eq123d.2 . . 3  |-  ( ph  ->  A  =  B )
5 f1oeq2 5544 . . 3  |-  ( A  =  B  ->  ( G : A -1-1-onto-> C  <->  G : B -1-1-onto-> C ) )
64, 5syl 15 . 2  |-  ( ph  ->  ( G : A -1-1-onto-> C  <->  G : B -1-1-onto-> C ) )
7 f1eq123d.3 . . 3  |-  ( ph  ->  C  =  D )
8 f1oeq3 5545 . . 3  |-  ( C  =  D  ->  ( G : B -1-1-onto-> C  <->  G : B -1-1-onto-> D ) )
97, 8syl 15 . 2  |-  ( ph  ->  ( G : B -1-1-onto-> C  <->  G : B -1-1-onto-> D ) )
103, 6, 93bitrd 270 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 176    = wceq 1642   -1-1-onto->wf1o 5333
This theorem is referenced by:  idffth  13900  ressffth  13905  indf1ofs  23689  f1oprg  27420  s2f1o  27494  s4f1o  27496
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4103  df-opab 4157  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341
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