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Theorem lcfl1 36307
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 508 . 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 36306 . 2  |-  ( G  e.  C  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G ) ) )
52, 4syl6rbbr 264 1  |-  ( ph  ->  ( G  e.  C  <->  ( 
._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596
This theorem is referenced by:  lcfl2  36308  lcfl5  36311  lcfl5a  36312  lcfl6  36315  lcfl8  36317  lcfl8a  36318  lclkrlem2  36347
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