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Theorem predel 25443
Description: Membership in the predecessor class implies membership in the base class. (Contributed by Scott Fenton, 11-Feb-2011.)
Assertion
Ref Expression
predel  |-  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y  e.  A )

Proof of Theorem predel
StepHypRef Expression
1 elin 3522 . . 3  |-  ( Y  e.  ( A  i^i  ( `' R " { X } ) )  <->  ( Y  e.  A  /\  Y  e.  ( `' R " { X } ) ) )
21simplbi 447 . 2  |-  ( Y  e.  ( A  i^i  ( `' R " { X } ) )  ->  Y  e.  A )
3 df-pred 25427 . 2  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
42, 3eleq2s 2527 1  |-  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    i^i cin 3311   {csn 3806   `'ccnv 4869   "cima 4873   Predcpred 25426
This theorem is referenced by:  predpo  25444  predpoirr  25457  predfrirr  25458  dftrpred3g  25496
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-pred 25427
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