Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  predeq3 Unicode version

Theorem predeq3 25195
Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
predeq3  |-  ( X  =  Y  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y ) )

Proof of Theorem predeq3
StepHypRef Expression
1 sneq 3768 . . . 4  |-  ( X  =  Y  ->  { X }  =  { Y } )
21imaeq2d 5143 . . 3  |-  ( X  =  Y  ->  ( `' R " { X } )  =  ( `' R " { Y } ) )
32ineq2d 3485 . 2  |-  ( X  =  Y  ->  ( A  i^i  ( `' R " { X } ) )  =  ( A  i^i  ( `' R " { Y } ) ) )
4 df-pred 25192 . 2  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
5 df-pred 25192 . 2  |-  Pred ( R ,  A ,  Y )  =  ( A  i^i  ( `' R " { Y } ) )
63, 4, 53eqtr4g 2444 1  |-  ( X  =  Y  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    i^i cin 3262   {csn 3757   `'ccnv 4817   "cima 4821   Predcpred 25191
This theorem is referenced by:  cbvsetlike  25205  predbrg  25210  preddowncl  25220  wfisg  25233  trpredeq3  25249  trpredlem1  25254  trpredtr  25257  trpredmintr  25258  trpredrec  25265  frmin  25266  frinsg  25269  wfr3g  25279  wfrlem1  25280  wfrlem9  25288  wfrlem14  25293  wfrlem15  25294  wfr2  25297  wfr2c  25298  tfr3ALT  25303  frr3g  25304  frrlem1  25305  frrlem5e  25313  bpolyval  25809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-xp 4824  df-cnv 4826  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-pred 25192
  Copyright terms: Public domain W3C validator