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Theorem mapdordlem1bN 32370
Description: Lemma for mapdord 32373. (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 32226 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 1652    e. wcel 1725   {crab 2701   ` cfv 5446
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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