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Theorem predeq3 24242
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 3664 . . . 4  |-  ( X  =  Y  ->  { X }  =  { Y } )
21imaeq2d 5028 . . 3  |-  ( X  =  Y  ->  ( `' R " { X } )  =  ( `' R " { Y } ) )
32ineq2d 3383 . 2  |-  ( X  =  Y  ->  ( A  i^i  ( `' R " { X } ) )  =  ( A  i^i  ( `' R " { Y } ) ) )
4 df-pred 24239 . 2  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
5 df-pred 24239 . 2  |-  Pred ( R ,  A ,  Y )  =  ( A  i^i  ( `' R " { Y } ) )
63, 4, 53eqtr4g 2353 1  |-  ( X  =  Y  ->  Pred ( R ,  A ,  X )  =  Pred ( R ,  A ,  Y ) )
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:  cbvsetlike  24252  predbrg  24257  preddowncl  24267  wfisg  24280  trpredeq3  24296  trpredlem1  24301  trpredtr  24304  trpredmintr  24305  trpredrec  24312  frmin  24313  frinsg  24316  wfr3g  24326  wfrlem1  24327  wfrlem9  24335  wfrlem14  24340  wfrlem15  24341  wfr2  24344  wfr2c  24345  tfr3ALT  24350  frr3g  24351  frrlem1  24352  frrlem5e  24360  bpolyval  24856
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-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-pred 24239
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