MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1eq123d Structured version   Unicode version

Theorem f1eq123d 5732
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 5697 . . 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 5698 . . 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 5699 . . 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 1399   -1-1->wf1 5506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-rab 2751  df-v 3049  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-sn 3958  df-pr 3960  df-op 3964  df-br 4381  df-opab 4439  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514
This theorem is referenced by:  fthf1  15342  cofth  15360  istrkgld  23993  istrkg2ld  23994  isushgra  24443  usgraeq12d  24504  usgra0v  24513  usgra1v  24532  usgrares1  24552  2spontn0vne  25029  aciunf1  27679  usgra2pthspth  32704  isushgr  32720  usgedgleord  32772  isfusgra  32777  usgresvm1  32796  usgresvm1ALT  32800
  Copyright terms: Public domain W3C validator