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Theorem lcfls1c 32348
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 936 . 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 32346 . 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 32303 . . 3  |-  ( G  e.  D  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G ) ) )
65anbi1i 676 . 2  |-  ( ( G  e.  D  /\  (  ._|_  `  ( L `  G ) )  C_  Q )  <->  ( ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G
) ) )  =  ( L `  G
) )  /\  (  ._|_  `  ( L `  G ) )  C_  Q ) )
71, 3, 63bitr4i 268 1  |-  ( G  e.  C  <->  ( G  e.  D  /\  (  ._|_  `  ( L `  G ) )  C_  Q ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {crab 2560    C_ wss 3165   ` cfv 5271
This theorem is referenced by:  lclkrslem1  32349  lclkrslem2  32350
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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