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Theorem foeq123d 5673
Description: Equality deduction for 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
foeq123d  |-  ( ph  ->  ( F : A -onto-> C 
<->  G : B -onto-> D
) )

Proof of Theorem foeq123d
StepHypRef Expression
1 f1eq123d.1 . . 3  |-  ( ph  ->  F  =  G )
2 foeq1 5652 . . 3  |-  ( F  =  G  ->  ( F : A -onto-> C  <->  G : A -onto-> C ) )
31, 2syl 16 . 2  |-  ( ph  ->  ( F : A -onto-> C 
<->  G : A -onto-> C
) )
4 f1eq123d.2 . . 3  |-  ( ph  ->  A  =  B )
5 foeq2 5653 . . 3  |-  ( A  =  B  ->  ( G : A -onto-> C  <->  G : B -onto-> C ) )
64, 5syl 16 . 2  |-  ( ph  ->  ( G : A -onto-> C 
<->  G : B -onto-> C
) )
7 f1eq123d.3 . . 3  |-  ( ph  ->  C  =  D )
8 foeq3 5654 . . 3  |-  ( C  =  D  ->  ( G : B -onto-> C  <->  G : B -onto-> D ) )
97, 8syl 16 . 2  |-  ( ph  ->  ( G : B -onto-> C 
<->  G : B -onto-> D
) )
103, 6, 93bitrd 272 1  |-  ( ph  ->  ( F : A -onto-> C 
<->  G : B -onto-> D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653   -onto->wfo 5455
This theorem is referenced by:  fullfo  14114  cofull  14136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-fun 5459  df-fn 5460  df-fo 5463
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