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Metamath Proof Explorer

Theorem dibfna

Description: Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014)

Ref Expression
Hypotheses dibfna.h 
|- H = ( LHyp ` K )
dibfna.j 
|- J = ( ( DIsoA ` K ) ` W )
dibfna.i 
|- I = ( ( DIsoB ` K ) ` W )
Assertion dibfna 
|- ( ( K e. V /\ W e. H ) -> I Fn dom J )

Proof

Step Hyp Ref Expression
1 dibfna.h 
 |-  H = ( LHyp ` K )
2 dibfna.j 
 |-  J = ( ( DIsoA ` K ) ` W )
3 dibfna.i 
 |-  I = ( ( DIsoB ` K ) ` W )
4 fvex 
 |-  ( J ` y ) e. _V
5 snex 
 |-  { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } e. _V
6 4 5 xpex 
 |-  ( ( J ` y ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) e. _V
7 eqid 
 |-  ( y e. dom J |-> ( ( J ` y ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) ) = ( y e. dom J |-> ( ( J ` y ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) )
8 6 7 fnmpti 
 |-  ( y e. dom J |-> ( ( J ` y ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) ) Fn dom J
9 eqid 
 |-  ( Base ` K ) = ( Base ` K )
10 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
11 eqid 
 |-  ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) )
12 9 1 10 11 2 3 dibfval 
 |-  ( ( K e. V /\ W e. H ) -> I = ( y e. dom J |-> ( ( J ` y ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) ) )
13 12 fneq1d 
 |-  ( ( K e. V /\ W e. H ) -> ( I Fn dom J <-> ( y e. dom J |-> ( ( J ` y ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) ) Fn dom J ) )
14 8 13 mpbiri 
 |-  ( ( K e. V /\ W e. H ) -> I Fn dom J )