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Theorem mapdordlem1 31826
Description: Lemma for mapdord 31828. (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 5525 . . . . 5  |-  ( g  =  J  ->  ( L `  g )  =  ( L `  J ) )
21fveq2d 5529 . . . 4  |-  ( g  =  J  ->  ( O `  ( L `  g ) )  =  ( O `  ( L `  J )
) )
32fveq2d 5529 . . 3  |-  ( g  =  J  ->  ( O `  ( O `  ( L `  g
) ) )  =  ( O `  ( O `  ( L `  J ) ) ) )
43eleq1d 2349 . 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 2925 1  |-  ( J  e.  T  <->  ( J  e.  F  /\  ( O `  ( O `  ( L `  J
) ) )  e.  Y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   ` cfv 5255
This theorem is referenced by:  mapdordlem2  31827
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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