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Theorem mapdordlem1 32361
Description: Lemma for mapdord 32363. (Contributed by NM, 27-Jan-2015.)
Hypothesis
Ref Expression
mapdordlem1.t  |-  T  =  { g  e.  F  |  ( O `  ( O `  ( L `
 g ) ) )  e.  Y }
Assertion
Ref Expression
mapdordlem1  |-  ( J  e.  T  <->  ( J  e.  F  /\  ( O `  ( O `  ( L `  J
) ) )  e.  Y ) )
Distinct variable groups:    g, F    g, J    g, L    g, O    g, Y
Allowed substitution hint:    T( g)

Proof of Theorem mapdordlem1
StepHypRef Expression
1 fveq2 5720 . . . . 5  |-  ( g  =  J  ->  ( L `  g )  =  ( L `  J ) )
21fveq2d 5724 . . . 4  |-  ( g  =  J  ->  ( O `  ( L `  g ) )  =  ( O `  ( L `  J )
) )
32fveq2d 5724 . . 3  |-  ( g  =  J  ->  ( O `  ( O `  ( L `  g
) ) )  =  ( O `  ( O `  ( L `  J ) ) ) )
43eleq1d 2501 . 2  |-  ( g  =  J  ->  (
( O `  ( O `  ( L `  g ) ) )  e.  Y  <->  ( O `  ( O `  ( L `  J )
) )  e.  Y
) )
5 mapdordlem1.t . 2  |-  T  =  { g  e.  F  |  ( O `  ( O `  ( L `
 g ) ) )  e.  Y }
64, 5elrab2 3086 1  |-  ( J  e.  T  <->  ( J  e.  F  /\  ( O `  ( O `  ( L `  J
) ) )  e.  Y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701   ` cfv 5446
This theorem is referenced by:  mapdordlem2  32362
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-3an 938  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-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454
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