Theorem lcfls1N 35148

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

Hypotheses

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

Assertion

Ref	Expression
lcfls1N	|- ( ph -> ( G e. C <-> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) ) )

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

Allowed substitution hints:   ph( f)   C( f)

Proof of Theorem lcfls1N

Step	Hyp	Ref	Expression
1		lcfls1.g	. . 3 |- ( ph -> G e. F )
2	1	biantrurd 515	. 2 |- ( ph -> ( ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) <-> ( G e. F /\ ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) ) ) )
3		lcfls1.c	. . . 4 |- C = { f e. F | ( ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) /\ ( ._|_ ` ( L ` f ) ) C_ Q ) }
4	3	lcfls1lem 35147	. . 3 |- ( G e. C <-> ( G e. F /\ ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) )
5		3anass 995	. . 3 |- ( ( G e. F /\ ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) <-> ( G e. F /\ ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) ) )
6	4, 5	bitri 257	. 2 |- ( G e. C <-> ( G e. F /\ ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) ) )
7	2, 6	syl6rbbr 272	1 |- ( ph -> ( G e. C <-> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   {crab 2753   C_ wss 3416   ` cfv 5601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-iota 5565  df-fv 5609

This theorem is referenced by: (None)