Theorem dihcnvid1

Description: The converse isomorphism of an isomorphism. (Contributed by NM, 5-Aug-2014)

	Ref	Expression
Hypotheses	dihcnvid1.b	|- B = ( Base ` K )
	dihcnvid1.h	|- H = ( LHyp ` K )
	dihcnvid1.i	|- I = ( ( DIsoH ` K ) ` W )
Assertion	dihcnvid1	|- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( `' I ` ( I ` X ) ) = X )

Proof

Step	Hyp	Ref	Expression
1		dihcnvid1.b	|- B = ( Base ` K )
2		dihcnvid1.h	|- H = ( LHyp ` K )
3		dihcnvid1.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		f1ocnvfv1	|- ( ( I : B -1-1-onto-> ran I /\ X e. B ) -> ( `' I ` ( I ` X ) ) = X )
10	8 9	sylan	|- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( `' I ` ( I ` X ) ) = X )