Theorem dihcnvcl

Description: Closure of isomorphism H converse. (Contributed by NM, 8-Mar-2014)

	Ref	Expression
Hypotheses	dihfn.b	|- B = ( Base ` K )
	dihfn.h	|- H = ( LHyp ` K )
	dihfn.i	|- I = ( ( DIsoH ` K ) ` W )
Assertion	dihcnvcl	|- ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( `' I ` X ) e. B )

Proof

Step	Hyp	Ref	Expression
1		dihfn.b	|- B = ( Base ` K )
2		dihfn.h	|- H = ( LHyp ` K )
3		dihfn.i	|- I = ( ( DIsoH ` K ) ` W )
4		eqid	|- ( ( DVecH ` K ) ` W ) = ( ( DVecH ` K ) ` W )
5		eqid	|- ( LSubSp ` ( ( DVecH ` K ) ` W ) ) = ( LSubSp ` ( ( DVecH ` K ) ` W ) )
6	1 2 3 4 5	dihf11	|- ( ( K e. HL /\ W e. H ) -> I : B -1-1-> ( LSubSp ` ( ( DVecH ` K ) ` W ) ) )
7		f1f1orn	|- ( I : B -1-1-> ( LSubSp ` ( ( DVecH ` K ) ` W ) ) -> I : B -1-1-onto-> ran I )
8	6 7	syl	|- ( ( K e. HL /\ W e. H ) -> I : B -1-1-onto-> ran I )
9		f1ocnvdm	|- ( ( I : B -1-1-onto-> ran I /\ X e. ran I ) -> ( `' I ` X ) e. B )
10	8 9	sylan	|- ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( `' I ` X ) e. B )