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Theorem foeq123d 5262
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 5242 . . 3  |-  ( F  =  G  ->  ( F : A -onto-> C  <->  G : A -onto-> C ) )
31, 2syl 14 . 2  |-  ( ph  ->  ( F : A -onto-> C 
<->  G : A -onto-> C
) )
4 f1eq123d.2 . . 3  |-  ( ph  ->  A  =  B )
5 foeq2 5243 . . 3  |-  ( A  =  B  ->  ( G : A -onto-> C  <->  G : B -onto-> C ) )
64, 5syl 14 . 2  |-  ( ph  ->  ( G : A -onto-> C 
<->  G : B -onto-> C
) )
7 f1eq123d.3 . . 3  |-  ( ph  ->  C  =  D )
8 foeq3 5244 . . 3  |-  ( C  =  D  ->  ( G : B -onto-> C  <->  G : B -onto-> D ) )
97, 8syl 14 . 2  |-  ( ph  ->  ( G : B -onto-> C 
<->  G : B -onto-> D
) )
103, 6, 93bitrd 213 1  |-  ( ph  ->  ( F : A -onto-> C 
<->  G : B -onto-> D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1290   -onto->wfo 5026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-fun 5030  df-fn 5031  df-fo 5034
This theorem is referenced by: (None)
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