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Theorem mapdordlem1 31752
Description: Lemma for mapdord 31754. (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 5669 . . . . 5  |-  ( g  =  J  ->  ( L `  g )  =  ( L `  J ) )
21fveq2d 5673 . . . 4  |-  ( g  =  J  ->  ( O `  ( L `  g ) )  =  ( O `  ( L `  J )
) )
32fveq2d 5673 . . 3  |-  ( g  =  J  ->  ( O `  ( O `  ( L `  g
) ) )  =  ( O `  ( O `  ( L `  J ) ) ) )
43eleq1d 2454 . 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 3038 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 1649    e. wcel 1717   {crab 2654   ` cfv 5395
This theorem is referenced by:  mapdordlem2  31753
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-3an 938  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-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403
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