Theorem dibvalrel

Description: The value of partial isomorphism B is a relation. (Contributed by NM, 8-Mar-2014)

	Ref	Expression
Hypotheses	dibcl.h	|- H = ( LHyp ` K )
	dibcl.i	|- I = ( ( DIsoB ` K ) ` W )
Assertion	dibvalrel	|- ( ( K e. V /\ W e. H ) -> Rel ( I ` X ) )

Proof

Step	Hyp	Ref	Expression
1		dibcl.h	|- H = ( LHyp ` K )
2		dibcl.i	|- I = ( ( DIsoB ` K ) ` W )
3		relxp	|- Rel ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } )
4		eqid	|- ( ( DIsoA ` K ) ` W ) = ( ( DIsoA ` K ) ` W )
5	1 4 2	dibdiadm	|- ( ( K e. V /\ W e. H ) -> dom I = dom ( ( DIsoA ` K ) ` W ) )
6	5	eleq2d	|- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> X e. dom ( ( DIsoA ` K ) ` W ) ) )
7	6	biimpa	|- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> X e. dom ( ( DIsoA ` K ) ` W ) )
8		eqid	|- ( Base ` K ) = ( Base ` K )
9		eqid	|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
10		eqid	|- ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) = ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) )
11	8 1 9 10 4 2	dibval	|- ( ( ( K e. V /\ W e. H ) /\ X e. dom ( ( DIsoA ` K ) ` W ) ) -> ( I ` X ) = ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) )
12	7 11	syldan	|- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> ( I ` X ) = ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) )
13	12	releqd	|- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> ( Rel ( I ` X ) <-> Rel ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) ) )
14	3 13	mpbiri	|- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> Rel ( I ` X ) )
15		rel0	|- Rel (/)
16		ndmfv	|- ( -. X e. dom I -> ( I ` X ) = (/) )
17	16	releqd	|- ( -. X e. dom I -> ( Rel ( I ` X ) <-> Rel (/) ) )
18	15 17	mpbiri	|- ( -. X e. dom I -> Rel ( I ` X ) )
19	18	adantl	|- ( ( ( K e. V /\ W e. H ) /\ -. X e. dom I ) -> Rel ( I ` X ) )
20	14 19	pm2.61dan	|- ( ( K e. V /\ W e. H ) -> Rel ( I ` X ) )