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Theorem foeq123d 5637
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 5616 . . 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 5617 . . 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 5618 . . 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 271 1  |-  ( ph  ->  ( F : A -onto-> C 
<->  G : B -onto-> D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649   -onto->wfo 5419
This theorem is referenced by:  fullfo  14072  cofull  14094
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 2393
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 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-fun 5423  df-fn 5424  df-fo 5427
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