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Theorem foeq123d 5551
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 5530 . . 3  |-  ( F  =  G  ->  ( F : A -onto-> C  <->  G : A -onto-> C ) )
31, 2syl 15 . 2  |-  ( ph  ->  ( F : A -onto-> C 
<->  G : A -onto-> C
) )
4 f1eq123d.2 . . 3  |-  ( ph  ->  A  =  B )
5 foeq2 5531 . . 3  |-  ( A  =  B  ->  ( G : A -onto-> C  <->  G : B -onto-> C ) )
64, 5syl 15 . 2  |-  ( ph  ->  ( G : A -onto-> C 
<->  G : B -onto-> C
) )
7 f1eq123d.3 . . 3  |-  ( ph  ->  C  =  D )
8 foeq3 5532 . . 3  |-  ( C  =  D  ->  ( G : B -onto-> C  <->  G : B -onto-> D ) )
97, 8syl 15 . 2  |-  ( ph  ->  ( G : B -onto-> C 
<->  G : B -onto-> D
) )
103, 6, 93bitrd 270 1  |-  ( ph  ->  ( F : A -onto-> C 
<->  G : B -onto-> D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1642   -onto->wfo 5335
This theorem is referenced by:  fullfo  13885  cofull  13907
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 4105  df-opab 4159  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-fun 5339  df-fn 5340  df-fo 5343
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