Theorem dibelval1st

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

	Ref	Expression
Hypotheses	dibelval1.b	|- B = ( Base ` K )
	dibelval1.l	|- .<_ = ( le ` K )
	dibelval1.h	|- H = ( LHyp ` K )
	dibelval1.j	|- J = ( ( DIsoA ` K ) ` W )
	dibelval1.i	|- I = ( ( DIsoB ` K ) ` W )
Assertion	dibelval1st	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 1st ` Y ) e. ( J ` X ) )

Proof

Step	Hyp	Ref	Expression
1		dibelval1.b	|- B = ( Base ` K )
2		dibelval1.l	|- .<_ = ( le ` K )
3		dibelval1.h	|- H = ( LHyp ` K )
4		dibelval1.j	|- J = ( ( DIsoA ` K ) ` W )
5		dibelval1.i	|- I = ( ( DIsoB ` K ) ` W )
6		eqid	|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
7		eqid	|- ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) )
8	1 2 3 6 7 4 5	dibval2	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( J ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) )
9	8	eleq2d	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( Y e. ( I ` X ) <-> Y e. ( ( J ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) ) )
10	9	biimp3a	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> Y e. ( ( J ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) )
11		xp1st	|- ( Y e. ( ( J ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) -> ( 1st ` Y ) e. ( J ` X ) )
12	10 11	syl	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 1st ` Y ) e. ( J ` X ) )