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Theorem f1eq123d 5467
Description: Equality deduction for one-to-one 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
f1eq123d  |-  ( ph  ->  ( F : A -1-1-> C  <-> 
G : B -1-1-> D
) )

Proof of Theorem f1eq123d
StepHypRef Expression
1 f1eq123d.1 . . 3  |-  ( ph  ->  F  =  G )
2 f1eq1 5432 . . 3  |-  ( F  =  G  ->  ( F : A -1-1-> C  <->  G : A -1-1-> C ) )
31, 2syl 15 . 2  |-  ( ph  ->  ( F : A -1-1-> C  <-> 
G : A -1-1-> C
) )
4 f1eq123d.2 . . 3  |-  ( ph  ->  A  =  B )
5 f1eq2 5433 . . 3  |-  ( A  =  B  ->  ( G : A -1-1-> C  <->  G : B -1-1-> C ) )
64, 5syl 15 . 2  |-  ( ph  ->  ( G : A -1-1-> C  <-> 
G : B -1-1-> C
) )
7 f1eq123d.3 . . 3  |-  ( ph  ->  C  =  D )
8 f1eq3 5434 . . 3  |-  ( C  =  D  ->  ( G : B -1-1-> C  <->  G : B -1-1-> D ) )
97, 8syl 15 . 2  |-  ( ph  ->  ( G : B -1-1-> C  <-> 
G : B -1-1-> D
) )
103, 6, 93bitrd 270 1  |-  ( ph  ->  ( F : A -1-1-> C  <-> 
G : B -1-1-> D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623   -1-1->wf1 5252
This theorem is referenced by:  fthf1  13791  cofth  13809  usgraeq12d  28111  usgra0v  28117  usgra1v  28126
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260
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