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Theorem elpred 29110
Description: Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.)
Hypothesis
Ref Expression
elpred.1  |-  Y  e. 
_V
Assertion
Ref Expression
elpred  |-  ( X  e.  D  ->  ( Y  e.  Pred ( R ,  A ,  X
)  <->  ( Y  e.  A  /\  Y R X ) ) )

Proof of Theorem elpred
StepHypRef Expression
1 df-pred 29097 . . 3  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
21elin2 3689 . 2  |-  ( Y  e.  Pred ( R ,  A ,  X )  <->  ( Y  e.  A  /\  Y  e.  ( `' R " { X }
) ) )
3 elpred.1 . . . 4  |-  Y  e. 
_V
43eliniseg 5366 . . 3  |-  ( X  e.  D  ->  ( Y  e.  ( `' R " { X }
)  <->  Y R X ) )
54anbi2d 703 . 2  |-  ( X  e.  D  ->  (
( Y  e.  A  /\  Y  e.  ( `' R " { X } ) )  <->  ( Y  e.  A  /\  Y R X ) ) )
62, 5syl5bb 257 1  |-  ( X  e.  D  ->  ( Y  e.  Pred ( R ,  A ,  X
)  <->  ( Y  e.  A  /\  Y R X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767   _Vcvv 3113   {csn 4027   class class class wbr 4447   `'ccnv 4998   "cima 5002   Predcpred 29096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-pred 29097
This theorem is referenced by:  predreseq  29112  predpo  29117  setlikespec  29120  preddowncl  29129  preduz  29133  predfz  29136  wfrlem10  29205  wzel  29233  wsuclem  29234
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