Theorem dibelval1st2N

Description: Membership in value of the partial isomorphism B for a lattice K . (Contributed by NM, 13-Feb-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dibelval1st2.b	|- B = ( Base ` K )
	dibelval1st2.l	|- .<_ = ( le ` K )
	dibelval1st2.h	|- H = ( LHyp ` K )
	dibelval1st2.t	|- T = ( ( LTrn ` K ) ` W )
	dibelval1st2.r	|- R = ( ( trL ` K ) ` W )
	dibelval1st2.i	|- I = ( ( DIsoB ` K ) ` W )
Assertion	dibelval1st2N	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( R ` ( 1st ` Y ) ) .<_ X )

Proof

Step	Hyp	Ref	Expression
1		dibelval1st2.b	|- B = ( Base ` K )
2		dibelval1st2.l	|- .<_ = ( le ` K )
3		dibelval1st2.h	|- H = ( LHyp ` K )
4		dibelval1st2.t	|- T = ( ( LTrn ` K ) ` W )
5		dibelval1st2.r	|- R = ( ( trL ` K ) ` W )
6		dibelval1st2.i	|- I = ( ( DIsoB ` K ) ` W )
7		eqid	|- ( ( DIsoA ` K ) ` W ) = ( ( DIsoA ` K ) ` W )
8	1 2 3 7 6	dibelval1st	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 1st ` Y ) e. ( ( ( DIsoA ` K ) ` W ) ` X ) )
9	1 2 3 4 5 7	diatrl	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( 1st ` Y ) e. ( ( ( DIsoA ` K ) ` W ) ` X ) ) -> ( R ` ( 1st ` Y ) ) .<_ X )
10	8 9	syld3an3	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( R ` ( 1st ` Y ) ) .<_ X )