Theorem dibelval1st1

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

	Ref	Expression
Hypotheses	dibelval1st1.b	|- B = ( Base ` K )
	dibelval1st1.l	|- .<_ = ( le ` K )
	dibelval1st1.h	|- H = ( LHyp ` K )
	dibelval1st1.t	|- T = ( ( LTrn ` K ) ` W )
	dibelval1st1.i	|- I = ( ( DIsoB ` K ) ` W )
Assertion	dibelval1st1	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 1st ` Y ) e. T )

Proof

Step	Hyp	Ref	Expression
1		dibelval1st1.b	|- B = ( Base ` K )
2		dibelval1st1.l	|- .<_ = ( le ` K )
3		dibelval1st1.h	|- H = ( LHyp ` K )
4		dibelval1st1.t	|- T = ( ( LTrn ` K ) ` W )
5		dibelval1st1.i	|- I = ( ( DIsoB ` K ) ` W )
6		eqid	|- ( ( DIsoA ` K ) ` W ) = ( ( DIsoA ` K ) ` W )
7	1 2 3 6 5	dibelval1st	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 1st ` Y ) e. ( ( ( DIsoA ` K ) ` W ) ` X ) )
8	1 2 3 4 6	diael	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( 1st ` Y ) e. ( ( ( DIsoA ` K ) ` W ) ` X ) ) -> ( 1st ` Y ) e. T )
9	7 8	syld3an3	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 1st ` Y ) e. T )