Theorem dibval2

Description: Value of the partial isomorphism B. (Contributed by NM, 18-Jan-2014)

	Ref	Expression
Hypotheses	dibval2.b	|- B = ( Base ` K )
	dibval2.l	|- .<_ = ( le ` K )
	dibval2.h	|- H = ( LHyp ` K )
	dibval2.t	|- T = ( ( LTrn ` K ) ` W )
	dibval2.o	|- .0. = ( f e. T |-> ( _I |` B ) )
	dibval2.j	|- J = ( ( DIsoA ` K ) ` W )
	dibval2.i	|- I = ( ( DIsoB ` K ) ` W )
Assertion	dibval2	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( J ` X ) X. { .0. } ) )

Proof

Step	Hyp	Ref	Expression
1		dibval2.b	|- B = ( Base ` K )
2		dibval2.l	|- .<_ = ( le ` K )
3		dibval2.h	|- H = ( LHyp ` K )
4		dibval2.t	|- T = ( ( LTrn ` K ) ` W )
5		dibval2.o	|- .0. = ( f e. T |-> ( _I |` B ) )
6		dibval2.j	|- J = ( ( DIsoA ` K ) ` W )
7		dibval2.i	|- I = ( ( DIsoB ` K ) ` W )
8	1 2 3 6	diaeldm	|- ( ( K e. V /\ W e. H ) -> ( X e. dom J <-> ( X e. B /\ X .<_ W ) ) )
9	8	biimpar	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> X e. dom J )
10	1 3 4 5 6 7	dibval	|- ( ( ( K e. V /\ W e. H ) /\ X e. dom J ) -> ( I ` X ) = ( ( J ` X ) X. { .0. } ) )
11	9 10	syldan	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( J ` X ) X. { .0. } ) )