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Theorem lcfl1 32304
Description: Property of a functional with a closed kernel. (Contributed by NM, 31-Dec-2014.)
Hypotheses
Ref Expression
lcfl1.c  |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
lcfl1.g  |-  ( ph  ->  G  e.  F )
Assertion
Ref Expression
lcfl1  |-  ( ph  ->  ( G  e.  C  <->  ( 
._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G ) ) )
Distinct variable groups:    f, F    f, G    f, L    ._|_ , f
Allowed substitution hints:    ph( f)    C( f)

Proof of Theorem lcfl1
StepHypRef Expression
1 lcfl1.g . . 3  |-  ( ph  ->  G  e.  F )
21biantrurd 494 . 2  |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G )  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G ) ) ) )
3 lcfl1.c . . 3  |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }
43lcfl1lem 32303 . 2  |-  ( G  e.  C  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G ) ) )
52, 4syl6rbbr 255 1  |-  ( ph  ->  ( G  e.  C  <->  ( 
._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560   ` cfv 5271
This theorem is referenced by:  lcfl2  32305  lcfl5  32308  lcfl5a  32309  lcfl6  32312  lcfl8  32314  lcfl8a  32315  lclkrlem2  32344
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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