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Theorem lcfl1lem 36163
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 5857 . . . . 5  |-  ( f  =  G  ->  ( L `  f )  =  ( L `  G ) )
21fveq2d 5861 . . . 4  |-  ( f  =  G  ->  (  ._|_  `  ( L `  f ) )  =  (  ._|_  `  ( L `
 G ) ) )
32fveq2d 5861 . . 3  |-  ( f  =  G  ->  (  ._|_  `  (  ._|_  `  ( L `  f )
) )  =  ( 
._|_  `  (  ._|_  `  ( L `  G )
) ) )
43, 1eqeq12d 2482 . 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 3256 1  |-  ( G  e.  C  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   {crab 2811   ` cfv 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-iota 5542  df-fv 5587
This theorem is referenced by:  lcfl1  36164  lcfl8b  36176  lclkrlem1  36178  lclkrlem2  36204  lclkr  36205  lcfls1c  36208  lcfrlem9  36222  mapdvalc  36301  mapdval2N  36302  mapdval4N  36304  mapdordlem1a  36306  mapdordlem1bN  36307  mapdrvallem2  36317
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