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Theorem foeq123d 5811
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 5790 . . 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 5791 . . 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 5792 . . 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 279 1  |-  ( ph  ->  ( F : A -onto-> C 
<->  G : B -onto-> D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379   -onto->wfo 5585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-fun 5589  df-fn 5590  df-fo 5593
This theorem is referenced by:  fullfo  15138  cofull  15160
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