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Metamath Proof Explorer

Theorem dibelval1st

Description: Membership in value of the partial isomorphism B for a lattice K . (Contributed by NM, 13-Feb-2014)

Ref Expression
Hypotheses dibelval1.b 
|- B = ( Base ` K )
dibelval1.l 
|- .<_ = ( le ` K )
dibelval1.h 
|- H = ( LHyp ` K )
dibelval1.j 
|- J = ( ( DIsoA ` K ) ` W )
dibelval1.i 
|- I = ( ( DIsoB ` K ) ` W )
Assertion dibelval1st 
|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 1st ` Y ) e. ( J ` X ) )

Proof

Step Hyp Ref Expression
1 dibelval1.b 
 |-  B = ( Base ` K )
2 dibelval1.l 
 |-  .<_ = ( le ` K )
3 dibelval1.h 
 |-  H = ( LHyp ` K )
4 dibelval1.j 
 |-  J = ( ( DIsoA ` K ) ` W )
5 dibelval1.i 
 |-  I = ( ( DIsoB ` K ) ` W )
6 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
7 eqid 
 |-  ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) )
8 1 2 3 6 7 4 5 dibval2 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( J ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) )
9 8 eleq2d 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( Y e. ( I ` X ) <-> Y e. ( ( J ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) ) )
10 9 biimp3a 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> Y e. ( ( J ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) )
11 xp1st 
 |-  ( Y e. ( ( J ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) -> ( 1st ` Y ) e. ( J ` X ) )
12 10 11 syl 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 1st ` Y ) e. ( J ` X ) )