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Theorem lcfls1c 31653
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 31651 . 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 31608 . . 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 1649    e. wcel 1717   {crab 2655    C_ wss 3265   ` cfv 5396
This theorem is referenced by:  lclkrslem1  31654  lclkrslem2  31655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-iota 5360  df-fv 5404
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