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Theorem f1eq123d 5802
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 5767 . . 3  |-  ( F  =  G  ->  ( F : A -1-1-> C  <->  G : A -1-1-> C ) )
31, 2syl 16 . 2  |-  ( ph  ->  ( F : A -1-1-> C  <-> 
G : A -1-1-> C
) )
4 f1eq123d.2 . . 3  |-  ( ph  ->  A  =  B )
5 f1eq2 5768 . . 3  |-  ( A  =  B  ->  ( G : A -1-1-> C  <->  G : B -1-1-> C ) )
64, 5syl 16 . 2  |-  ( ph  ->  ( G : A -1-1-> C  <-> 
G : B -1-1-> C
) )
7 f1eq123d.3 . . 3  |-  ( ph  ->  C  =  D )
8 f1eq3 5769 . . 3  |-  ( C  =  D  ->  ( G : B -1-1-> C  <->  G : B -1-1-> D ) )
97, 8syl 16 . 2  |-  ( ph  ->  ( G : B -1-1-> C  <-> 
G : B -1-1-> D
) )
103, 6, 93bitrd 279 1  |-  ( ph  ->  ( F : A -1-1-> C  <-> 
G : B -1-1-> D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374   -1-1->wf1 5576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584
This theorem is referenced by:  fthf1  15133  cofth  15151  istrkgld  23578  istrkg2ld  23579  usgraeq12d  24025  usgra0v  24033  usgra1v  24052  usgrares1  24072  2spontn0vne  24549  usgra2pthspth  31639  usgedgleord  31685  isfusgra  31690  usgresvm1  31701
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