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Theorem predel 25208
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 3474 . . 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 25193 . 2  |-  Pred ( R ,  A ,  X )  =  ( A  i^i  ( `' R " { X } ) )
42, 3eleq2s 2480 1  |-  ( Y  e.  Pred ( R ,  A ,  X )  ->  Y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717    i^i cin 3263   {csn 3758   `'ccnv 4818   "cima 4822   Predcpred 25192
This theorem is referenced by:  predpo  25209  predpoirr  25222  predfrirr  25223  dftrpred3g  25261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902  df-in 3271  df-pred 25193
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