Theorem lcfl1 36781

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

Step	Hyp	Ref	Expression
1		lcfl1.g	. . 3 |- ( ph -> G e. F )
2	1	biantrurd 529	. 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 ) }
4	3	lcfl1lem 36780	. 2 |- ( G e. C <-> ( G e. F /\ ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) ) )
5	2, 4	syl6rbbr 279	1 |- ( ph -> ( G e. C <-> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  lcfl2  36782  lcfl5  36785  lcfl5a  36786  lcfl6  36789  lcfl8  36791  lcfl8a  36792  lclkrlem2  36821