Theorem lcfl1lem 36780

Description: Property of a functional with a closed kernel. (Contributed by NM, 28-Dec-2014.)

Hypothesis

Ref	Expression
lcfl1.c	|- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }

Assertion

Ref	Expression
lcfl1lem	|- ( G e. C <-> ( G e. F /\ ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) ) )

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

Allowed substitution hint:   C( f)

Proof of Theorem lcfl1lem

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 |- ( f = G -> ( L ` f ) = ( L ` G ) )
2	1	fveq2d 6195	. . . 4 |- ( f = G -> ( ._|_ ` ( L ` f ) ) = ( ._|_ ` ( L ` G ) ) )
3	2	fveq2d 6195	. . 3 |- ( f = G -> ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) )
4	3, 1	eqeq12d 2637	. 2 |- ( f = G -> ( ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) <-> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) ) )
5		lcfl1.c	. 2 |- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
6	4, 5	elrab2 3366	1 |- ( G e. C <-> ( G e. F /\ ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) ) )

Syntax hints:   <-> 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:  lcfl1  36781  lcfl8b  36793  lclkrlem1  36795  lclkrlem2  36821  lclkr  36822  lcfls1c  36825  lcfrlem9  36839  mapdvalc  36918  mapdval2N  36919  mapdval4N  36921  mapdordlem1a  36923  mapdordlem1bN  36924  mapdrvallem2  36934