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Theorem mapdordlem1 32448
Description: Lemma for mapdord 32450. (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 5541 . . . . 5  |-  ( g  =  J  ->  ( L `  g )  =  ( L `  J ) )
21fveq2d 5545 . . . 4  |-  ( g  =  J  ->  ( O `  ( L `  g ) )  =  ( O `  ( L `  J )
) )
32fveq2d 5545 . . 3  |-  ( g  =  J  ->  ( O `  ( O `  ( L `  g
) ) )  =  ( O `  ( O `  ( L `  J ) ) ) )
43eleq1d 2362 . 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 2938 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 1632    e. wcel 1696   {crab 2560   ` cfv 5271
This theorem is referenced by:  mapdordlem2  32449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279
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