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Theorem lcfl1lem 34511
Description: Property of a functional with a closed kernel. (Contributed by NM, 28-Dec-2014.)
Hypothesis
Ref Expression
lcfl1.c  |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
Assertion
Ref Expression
lcfl1lem  |-  ( G  e.  C  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G ) ) )
Distinct variable groups:    f, F    f, G    f, L    ._|_ , f
Allowed substitution hint:    C( f)

Proof of Theorem lcfl1lem
StepHypRef Expression
1 fveq2 5849 . . . . 5  |-  ( f  =  G  ->  ( L `  f )  =  ( L `  G ) )
21fveq2d 5853 . . . 4  |-  ( f  =  G  ->  (  ._|_  `  ( L `  f ) )  =  (  ._|_  `  ( L `
 G ) ) )
32fveq2d 5853 . . 3  |-  ( f  =  G  ->  (  ._|_  `  (  ._|_  `  ( L `  f )
) )  =  ( 
._|_  `  (  ._|_  `  ( L `  G )
) ) )
43, 1eqeq12d 2424 . 2  |-  ( f  =  G  ->  (
(  ._|_  `  (  ._|_  `  ( L `  f
) ) )  =  ( L `  f
)  <->  (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G ) ) )
5 lcfl1.c . 2  |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
64, 5elrab2 3209 1  |-  ( G  e.  C  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   {crab 2758   ` cfv 5569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577
This theorem is referenced by:  lcfl1  34512  lcfl8b  34524  lclkrlem1  34526  lclkrlem2  34552  lclkr  34553  lcfls1c  34556  lcfrlem9  34570  mapdvalc  34649  mapdval2N  34650  mapdval4N  34652  mapdordlem1a  34654  mapdordlem1bN  34655  mapdrvallem2  34665
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