Theorem mapdsn2

Description: Value of the map defined by df-mapd at the span of a singleton. (Contributed by NM, 16-Feb-2015)

	Ref	Expression
Hypotheses	mapdsn.h	|- H = ( LHyp ` K )
	mapdsn.o	|- O = ( ( ocH ` K ) ` W )
	mapdsn.m	|- M = ( ( mapd ` K ) ` W )
	mapdsn.u	|- U = ( ( DVecH ` K ) ` W )
	mapdsn.v	|- V = ( Base ` U )
	mapdsn.n	|- N = ( LSpan ` U )
	mapdsn.f	|- F = ( LFnl ` U )
	mapdsn.l	|- L = ( LKer ` U )
	mapdsn.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	mapdsn.x	|- ( ph -> X e. V )
	mapdsn2.e	|- ( ph -> ( L ` G ) = ( O ` { X } ) )
Assertion	mapdsn2	|- ( ph -> ( M ` ( N ` { X } ) ) = { f e. F | ( L ` G ) C_ ( L ` f ) } )

Proof

Step	Hyp	Ref	Expression
1		mapdsn.h	|- H = ( LHyp ` K )
2		mapdsn.o	|- O = ( ( ocH ` K ) ` W )
3		mapdsn.m	|- M = ( ( mapd ` K ) ` W )
4		mapdsn.u	|- U = ( ( DVecH ` K ) ` W )
5		mapdsn.v	|- V = ( Base ` U )
6		mapdsn.n	|- N = ( LSpan ` U )
7		mapdsn.f	|- F = ( LFnl ` U )
8		mapdsn.l	|- L = ( LKer ` U )
9		mapdsn.k	|- ( ph -> ( K e. HL /\ W e. H ) )
10		mapdsn.x	|- ( ph -> X e. V )
11		mapdsn2.e	|- ( ph -> ( L ` G ) = ( O ` { X } ) )
12	1 2 3 4 5 6 7 8 9 10	mapdsn	|- ( ph -> ( M ` ( N ` { X } ) ) = { f e. F | ( O ` { X } ) C_ ( L ` f ) } )
13	11	sseq1d	|- ( ph -> ( ( L ` G ) C_ ( L ` f ) <-> ( O ` { X } ) C_ ( L ` f ) ) )
14	13	rabbidv	|- ( ph -> { f e. F | ( L ` G ) C_ ( L ` f ) } = { f e. F | ( O ` { X } ) C_ ( L ` f ) } )
15	12 14	eqtr4d	|- ( ph -> ( M ` ( N ` { X } ) ) = { f e. F | ( L ` G ) C_ ( L ` f ) } )