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Theorem predeq1 24240
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 4871 . . . 4  |-  ( R  =  S  ->  `' R  =  `' S
)
21imaeq1d 5027 . . 3  |-  ( R  =  S  ->  ( `' R " { X } )  =  ( `' S " { X } ) )
32ineq2d 3383 . 2  |-  ( R  =  S  ->  ( A  i^i  ( `' R " { X } ) )  =  ( A  i^i  ( `' S " { X } ) ) )
4 df-pred 24239 . 2  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
5 df-pred 24239 . 2  |-  Pred ( S ,  A ,  X )  =  ( A  i^i  ( `' S " { X } ) )
63, 4, 53eqtr4g 2353 1  |-  ( R  =  S  ->  Pred ( R ,  A ,  X )  =  Pred ( S ,  A ,  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    i^i cin 3164   {csn 3653   `'ccnv 4704   "cima 4708   Predcpred 24238
This theorem is referenced by:  trpredeq1  24294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-pred 24239
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