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Theorem elpredim 24247
Description: Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.)
Hypothesis
Ref Expression
elpredim.1  |-  X  e. 
_V
Assertion
Ref Expression
elpredim  |-  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y R X )

Proof of Theorem elpredim
StepHypRef Expression
1 df-pred 24239 . . 3  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
21elin2 3372 . 2  |-  ( Y  e.  Pred ( R ,  A ,  X )  <->  ( Y  e.  A  /\  Y  e.  ( `' R " { X }
) ) )
3 elpredim.1 . . . . . 6  |-  X  e. 
_V
4 elimasng 5055 . . . . . . 7  |-  ( ( X  e.  _V  /\  Y  e.  ( `' R " { X }
) )  ->  ( Y  e.  ( `' R " { X }
)  <->  <. X ,  Y >.  e.  `' R ) )
5 opelcnvg 4877 . . . . . . 7  |-  ( ( X  e.  _V  /\  Y  e.  ( `' R " { X }
) )  ->  ( <. X ,  Y >.  e.  `' R  <->  <. Y ,  X >.  e.  R ) )
64, 5bitrd 244 . . . . . 6  |-  ( ( X  e.  _V  /\  Y  e.  ( `' R " { X }
) )  ->  ( Y  e.  ( `' R " { X }
)  <->  <. Y ,  X >.  e.  R ) )
73, 6mpan 651 . . . . 5  |-  ( Y  e.  ( `' R " { X } )  ->  ( Y  e.  ( `' R " { X } )  <->  <. Y ,  X >.  e.  R ) )
87ibi 232 . . . 4  |-  ( Y  e.  ( `' R " { X } )  ->  <. Y ,  X >.  e.  R )
9 df-br 4040 . . . 4  |-  ( Y R X  <->  <. Y ,  X >.  e.  R )
108, 9sylibr 203 . . 3  |-  ( Y  e.  ( `' R " { X } )  ->  Y R X )
1110adantl 452 . 2  |-  ( ( Y  e.  A  /\  Y  e.  ( `' R " { X }
) )  ->  Y R X )
122, 11sylbi 187 1  |-  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y R X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   _Vcvv 2801   {csn 3653   <.cop 3656   class class class wbr 4039   `'ccnv 4704   "cima 4708   Predcpred 24238
This theorem is referenced by:  predbrg  24257  preddowncl  24267  trpredrec  24312
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  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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