Theorem diaeldm

Description: Member of domain of the partial isomorphism A. (Contributed by NM, 4-Dec-2013)

	Ref	Expression
Hypotheses	diafn.b	|- B = ( Base ` K )
	diafn.l	|- .<_ = ( le ` K )
	diafn.h	|- H = ( LHyp ` K )
	diafn.i	|- I = ( ( DIsoA ` K ) ` W )
Assertion	diaeldm	|- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> ( X e. B /\ X .<_ W ) ) )

Proof

Step	Hyp	Ref	Expression
1		diafn.b	|- B = ( Base ` K )
2		diafn.l	|- .<_ = ( le ` K )
3		diafn.h	|- H = ( LHyp ` K )
4		diafn.i	|- I = ( ( DIsoA ` K ) ` W )
5	1 2 3 4	diadm	|- ( ( K e. V /\ W e. H ) -> dom I = { x e. B | x .<_ W } )
6	5	eleq2d	|- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> X e. { x e. B | x .<_ W } ) )
7		breq1	|- ( x = X -> ( x .<_ W <-> X .<_ W ) )
8	7	elrab	|- ( X e. { x e. B | x .<_ W } <-> ( X e. B /\ X .<_ W ) )
9	6 8	bitrdi	|- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> ( X e. B /\ X .<_ W ) ) )