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Theorem lcfls1lem 31724
Description: Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.)
Hypothesis
Ref Expression
lcfls1.c  |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f )  /\  (  ._|_  `  ( L `  f ) )  C_  Q ) }
Assertion
Ref Expression
lcfls1lem  |-  ( G  e.  C  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G )  /\  (  ._|_  `  ( L `  G )
)  C_  Q )
)
Distinct variable groups:    f, F    f, G    f, L    ._|_ , f    Q, f
Allowed substitution hint:    C( f)

Proof of Theorem lcfls1lem
StepHypRef Expression
1 fveq2 5525 . . . . . . 7  |-  ( f  =  G  ->  ( L `  f )  =  ( L `  G ) )
21fveq2d 5529 . . . . . 6  |-  ( f  =  G  ->  (  ._|_  `  ( L `  f ) )  =  (  ._|_  `  ( L `
 G ) ) )
32fveq2d 5529 . . . . 5  |-  ( f  =  G  ->  (  ._|_  `  (  ._|_  `  ( L `  f )
) )  =  ( 
._|_  `  (  ._|_  `  ( L `  G )
) ) )
43, 1eqeq12d 2297 . . . 4  |-  ( f  =  G  ->  (
(  ._|_  `  (  ._|_  `  ( L `  f
) ) )  =  ( L `  f
)  <->  (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G ) ) )
52sseq1d 3205 . . . 4  |-  ( f  =  G  ->  (
(  ._|_  `  ( L `  f ) )  C_  Q 
<->  (  ._|_  `  ( L `
 G ) ) 
C_  Q ) )
64, 5anbi12d 691 . . 3  |-  ( f  =  G  ->  (
( (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f )  /\  (  ._|_  `  ( L `  f ) )  C_  Q )  <->  ( (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G )  /\  (  ._|_  `  ( L `  G )
)  C_  Q )
) )
7 lcfls1.c . . 3  |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f )  /\  (  ._|_  `  ( L `  f ) )  C_  Q ) }
86, 7elrab2 2925 . 2  |-  ( G  e.  C  <->  ( G  e.  F  /\  (
(  ._|_  `  (  ._|_  `  ( L `  G
) ) )  =  ( L `  G
)  /\  (  ._|_  `  ( L `  G
) )  C_  Q
) ) )
9 3anass 938 . 2  |-  ( ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G
) ) )  =  ( L `  G
)  /\  (  ._|_  `  ( L `  G
) )  C_  Q
)  <->  ( G  e.  F  /\  ( ( 
._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G )  /\  (  ._|_  `  ( L `  G )
)  C_  Q )
) )
108, 9bitr4i 243 1  |-  ( G  e.  C  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G )  /\  (  ._|_  `  ( L `  G )
)  C_  Q )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   ` cfv 5255
This theorem is referenced by:  lcfls1N  31725  lcfls1c  31726  lclkrslem1  31727  lclkrslem2  31728  lclkrs  31729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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