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Theorem mapdordlem1bN 31752
Description: Lemma for mapdord 31755. (Contributed by NM, 27-Jan-2015.) (New usage is discouraged.)
Hypothesis
Ref Expression
mapdordlem1b.c  |-  C  =  { g  e.  F  |  ( O `  ( O `  ( L `
 g ) ) )  =  ( L `
 g ) }
Assertion
Ref Expression
mapdordlem1bN  |-  ( J  e.  C  <->  ( J  e.  F  /\  ( O `  ( O `  ( L `  J
) ) )  =  ( L `  J
) ) )
Distinct variable groups:    g, F    g, J    g, L    g, O
Allowed substitution hint:    C( g)

Proof of Theorem mapdordlem1bN
StepHypRef Expression
1 mapdordlem1b.c . 2  |-  C  =  { g  e.  F  |  ( O `  ( O `  ( L `
 g ) ) )  =  ( L `
 g ) }
21lcfl1lem 31608 1  |-  ( J  e.  C  <->  ( J  e.  F  /\  ( O `  ( O `  ( L `  J
) ) )  =  ( L `  J
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2655   ` cfv 5396
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 2370
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 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-iota 5360  df-fv 5404
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