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Theorem lcfl1lem 32303
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 5541 . . . . 5  |-  ( f  =  G  ->  ( L `  f )  =  ( L `  G ) )
21fveq2d 5545 . . . 4  |-  ( f  =  G  ->  (  ._|_  `  ( L `  f ) )  =  (  ._|_  `  ( L `
 G ) ) )
32fveq2d 5545 . . 3  |-  ( f  =  G  ->  (  ._|_  `  (  ._|_  `  ( L `  f )
) )  =  ( 
._|_  `  (  ._|_  `  ( L `  G )
) ) )
43, 1eqeq12d 2310 . 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 2938 1  |-  ( G  e.  C  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   ` cfv 5271
This theorem is referenced by:  lcfl1  32304  lcfl8b  32316  lclkrlem1  32318  lclkrlem2  32344  lclkr  32345  lcfls1c  32348  lcfrlem9  32362  mapdvalc  32441  mapdval2N  32442  mapdval4N  32444  mapdordlem1a  32446  mapdordlem1bN  32447  mapdrvallem2  32457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279
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