Theorem lcfls1c 36825

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

Step	Hyp	Ref	Expression
1		df-3an 1039	. 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 ) }
3	2	lcfls1lem 36823	. 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 ) }
5	4	lcfl1lem 36780	. . 3 |- ( G e. D <-> ( G e. F /\ ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) ) )
6	5	anbi1i 731	. 2 |- ( ( G e. D /\ ( ._|_ ` ( L ` G ) ) C_ Q ) <-> ( ( G e. F /\ ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) )
7	1, 3, 6	3bitr4i 292	1 |- ( G e. C <-> ( G e. D /\ ( ._|_ ` ( 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:  lclkrslem1  36826  lclkrslem2  36827