Theorem dibeldmN

Description: Member of domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		dibfn.b	|- B = ( Base ` K )
2		dibfn.l	|- .<_ = ( le ` K )
3		dibfn.h	|- H = ( LHyp ` K )
4		dibfn.i	|- I = ( ( DIsoB ` K ) ` W )
5		eqid	|- ( ( DIsoA ` K ) ` W ) = ( ( DIsoA ` K ) ` W )
6	3 5 4	dibdiadm	|- ( ( K e. V /\ W e. H ) -> dom I = dom ( ( DIsoA ` K ) ` W ) )
7	6	eleq2d	|- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> X e. dom ( ( DIsoA ` K ) ` W ) ) )
8	1 2 3 5	diaeldm	|- ( ( K e. V /\ W e. H ) -> ( X e. dom ( ( DIsoA ` K ) ` W ) <-> ( X e. B /\ X .<_ W ) ) )
9	7 8	bitrd	|- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> ( X e. B /\ X .<_ W ) ) )