This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )