Theorem lcfls1lem 36823

Description: Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.)

Hypothesis

Ref	Expression
lcfls1.c	|- C = { f e. F | ( ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) /\ ( ._|_ ` ( L ` f ) ) C_ Q ) }

Assertion

Ref	Expression
lcfls1lem	|- ( G e. C <-> ( G e. F /\ ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) )

Distinct variable groups:   f, F   f, G   f, L   ._|_ , f   Q, f

Allowed substitution hint:   C( f)

Proof of Theorem lcfls1lem

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . 7 |- ( f = G -> ( L ` f ) = ( L ` G ) )
2	1	fveq2d 6195	. . . . . 6 |- ( f = G -> ( ._|_ ` ( L ` f ) ) = ( ._|_ ` ( L ` G ) ) )
3	2	fveq2d 6195	. . . . 5 |- ( f = G -> ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) )
4	3, 1	eqeq12d 2637	. . . 4 |- ( f = G -> ( ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) <-> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) ) )
5	2	sseq1d 3632	. . . 4 |- ( f = G -> ( ( ._|_ ` ( L ` f ) ) C_ Q <-> ( ._|_ ` ( L ` G ) ) C_ Q ) )
6	4, 5	anbi12d 747	. . 3 |- ( f = G -> ( ( ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) /\ ( ._|_ ` ( L ` f ) ) C_ Q ) <-> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) ) )
7		lcfls1.c	. . 3 |- C = { f e. F | ( ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) /\ ( ._|_ ` ( L ` f ) ) C_ Q ) }
8	6, 7	elrab2 3366	. 2 |- ( G e. C <-> ( G e. F /\ ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) ) )
9		3anass 1042	. 2 |- ( ( G e. F /\ ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) <-> ( G e. F /\ ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) ) )
10	8, 9	bitr4i 267	1 |- ( G e. C <-> ( G e. F /\ ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   ` 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:  lcfls1N  36824  lcfls1c  36825  lclkrslem1  36826  lclkrslem2  36827  lclkrs  36828