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Theorem predeq3 24171
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 3651 . . . 4  |-  ( X  =  Y  ->  { X }  =  { Y } )
21imaeq2d 5012 . . 3  |-  ( X  =  Y  ->  ( `' R " { X } )  =  ( `' R " { Y } ) )
32ineq2d 3370 . 2  |-  ( X  =  Y  ->  ( A  i^i  ( `' R " { X } ) )  =  ( A  i^i  ( `' R " { Y } ) ) )
4 df-pred 24168 . 2  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
5 df-pred 24168 . 2  |-  Pred ( R ,  A ,  Y )  =  ( A  i^i  ( `' R " { Y } ) )
63, 4, 53eqtr4g 2340 1  |-  ( X  =  Y  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y ) )
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:  cbvsetlike  24181  predbrg  24186  preddowncl  24196  wfisg  24209  trpredeq3  24225  trpredlem1  24230  trpredtr  24233  trpredmintr  24234  trpredrec  24241  frmin  24242  frinsg  24245  wfr3g  24255  wfrlem1  24256  wfrlem9  24264  wfrlem14  24269  wfrlem15  24270  wfr2  24273  wfr2c  24274  tfr3ALT  24279  frr3g  24280  frrlem1  24281  frrlem5e  24289  bpolyval  24784
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-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-pred 24168
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