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Theorem f1oeq123d 5362
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 5356 . . 3  |-  ( F  =  G  ->  ( F : A -1-1-onto-> C  <->  G : A -1-1-onto-> C ) )
31, 2syl 14 . 2  |-  ( ph  ->  ( F : A -1-1-onto-> C  <->  G : A -1-1-onto-> C ) )
4 f1eq123d.2 . . 3  |-  ( ph  ->  A  =  B )
5 f1oeq2 5357 . . 3  |-  ( A  =  B  ->  ( G : A -1-1-onto-> C  <->  G : B -1-1-onto-> C ) )
64, 5syl 14 . 2  |-  ( ph  ->  ( G : A -1-1-onto-> C  <->  G : B -1-1-onto-> C ) )
7 f1eq123d.3 . . 3  |-  ( ph  ->  C  =  D )
8 f1oeq3 5358 . . 3  |-  ( C  =  D  ->  ( G : B -1-1-onto-> C  <->  G : B -1-1-onto-> D ) )
97, 8syl 14 . 2  |-  ( ph  ->  ( G : B -1-1-onto-> C  <->  G : B -1-1-onto-> D ) )
103, 6, 93bitrd 213 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 104    = wceq 1331   -1-1-onto->wf1o 5122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130
This theorem is referenced by:  f1oprg  5411  ennnfonelemhf1o  11926
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