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Theorem f1eq123d 5797
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 5762 . . 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 5763 . . 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 5764 . . 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 1381   -1-1->wf1 5571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-br 4434  df-opab 4492  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579
This theorem is referenced by:  fthf1  15155  cofth  15173  istrkgld  23722  istrkg2ld  23723  isushgra  24166  usgraeq12d  24227  usgra0v  24236  usgra1v  24255  usgrares1  24275  2spontn0vne  24752  usgra2pthspth  32185  isushgr  32201  usgedgleord  32253  isfusgra  32258  usgresvm1  32277  usgresvm1ALT  32281
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