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Theorem lcfls1c 32235
Description: Property of a functional with a closed kernel. (Contributed by NM, 28-Jan-2015.)
Hypotheses
Ref Expression
lcfls1.c  |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f )  /\  (  ._|_  `  ( L `  f ) )  C_  Q ) }
lcfls1c.c  |-  D  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
Assertion
Ref Expression
lcfls1c  |-  ( G  e.  C  <->  ( G  e.  D  /\  (  ._|_  `  ( L `  G ) )  C_  Q ) )
Distinct variable groups:    f, F    f, G    f, L    ._|_ , f    Q, f
Allowed substitution hints:    C( f)    D( f)

Proof of Theorem lcfls1c
StepHypRef Expression
1 df-3an 938 . 2  |-  ( ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G
) ) )  =  ( L `  G
)  /\  (  ._|_  `  ( L `  G
) )  C_  Q
)  <->  ( ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G ) )  /\  (  ._|_  `  ( L `  G
) )  C_  Q
) )
2 lcfls1.c . . 3  |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f )  /\  (  ._|_  `  ( L `  f ) )  C_  Q ) }
32lcfls1lem 32233 . 2  |-  ( G  e.  C  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G )  /\  (  ._|_  `  ( L `  G )
)  C_  Q )
)
4 lcfls1c.c . . . 4  |-  D  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
54lcfl1lem 32190 . . 3  |-  ( G  e.  D  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G ) ) )
65anbi1i 677 . 2  |-  ( ( G  e.  D  /\  (  ._|_  `  ( L `  G ) )  C_  Q )  <->  ( ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G
) ) )  =  ( L `  G
) )  /\  (  ._|_  `  ( L `  G ) )  C_  Q ) )
71, 3, 63bitr4i 269 1  |-  ( G  e.  C  <->  ( G  e.  D  /\  (  ._|_  `  ( L `  G ) )  C_  Q ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2701    C_ wss 3312   ` cfv 5446
This theorem is referenced by:  lclkrslem1  32236  lclkrslem2  32237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454
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