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Metamath Proof Explorer

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. } ) )