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Theorem lcfl1lem 35022
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 5879 . . . . 5  |-  ( f  =  G  ->  ( L `  f )  =  ( L `  G ) )
21fveq2d 5883 . . . 4  |-  ( f  =  G  ->  (  ._|_  `  ( L `  f ) )  =  (  ._|_  `  ( L `
 G ) ) )
32fveq2d 5883 . . 3  |-  ( f  =  G  ->  (  ._|_  `  (  ._|_  `  ( L `  f )
) )  =  ( 
._|_  `  (  ._|_  `  ( L `  G )
) ) )
43, 1eqeq12d 2445 . 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 3232 1  |-  ( G  e.  C  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1438    e. wcel 1869   {crab 2780   ` cfv 5599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-iota 5563  df-fv 5607
This theorem is referenced by:  lcfl1  35023  lcfl8b  35035  lclkrlem1  35037  lclkrlem2  35063  lclkr  35064  lcfls1c  35067  lcfrlem9  35081  mapdvalc  35160  mapdval2N  35161  mapdval4N  35163  mapdordlem1a  35165  mapdordlem1bN  35166  mapdrvallem2  35176
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