Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  predeq1 Unicode version

Theorem predeq1 24169
Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
predeq1  |-  ( R  =  S  ->  Pred ( R ,  A ,  X )  =  Pred ( S ,  A ,  X ) )

Proof of Theorem predeq1
StepHypRef Expression
1 cnveq 4855 . . . 4  |-  ( R  =  S  ->  `' R  =  `' S
)
21imaeq1d 5011 . . 3  |-  ( R  =  S  ->  ( `' R " { X } )  =  ( `' S " { X } ) )
32ineq2d 3370 . 2  |-  ( R  =  S  ->  ( A  i^i  ( `' R " { X } ) )  =  ( A  i^i  ( `' S " { X } ) ) )
4 df-pred 24168 . 2  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
5 df-pred 24168 . 2  |-  Pred ( S ,  A ,  X )  =  ( A  i^i  ( `' S " { X } ) )
63, 4, 53eqtr4g 2340 1  |-  ( R  =  S  ->  Pred ( R ,  A ,  X )  =  Pred ( S ,  A ,  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    i^i cin 3151   {csn 3640   `'ccnv 4688   "cima 4692   Predcpred 24167
This theorem is referenced by:  trpredeq1  24223
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-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-pred 24168
  Copyright terms: Public domain W3C validator