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Metamath Proof Explorer

Theorem dicval2

Description: The partial isomorphism C for a lattice K . (Contributed by NM, 20-Feb-2014)

Ref Expression
Hypotheses dicval.l 
|- .<_ = ( le ` K )
dicval.a 
|- A = ( Atoms ` K )
dicval.h 
|- H = ( LHyp ` K )
dicval.p 
|- P = ( ( oc ` K ) ` W )
dicval.t 
|- T = ( ( LTrn ` K ) ` W )
dicval.e 
|- E = ( ( TEndo ` K ) ` W )
dicval.i 
|- I = ( ( DIsoC ` K ) ` W )
dicval2.g 
|- G = ( iota_ g e. T ( g ` P ) = Q )
Assertion dicval2 
|- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) = { <. f , s >. | ( f = ( s ` G ) /\ s e. E ) } )

Proof

Step Hyp Ref Expression
1 dicval.l 
 |-  .<_ = ( le ` K )
2 dicval.a 
 |-  A = ( Atoms ` K )
3 dicval.h 
 |-  H = ( LHyp ` K )
4 dicval.p 
 |-  P = ( ( oc ` K ) ` W )
5 dicval.t 
 |-  T = ( ( LTrn ` K ) ` W )
6 dicval.e 
 |-  E = ( ( TEndo ` K ) ` W )
7 dicval.i 
 |-  I = ( ( DIsoC ` K ) ` W )
8 dicval2.g 
 |-  G = ( iota_ g e. T ( g ` P ) = Q )
9 1 2 3 4 5 6 7 dicval 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) = { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) } )
10 8 fveq2i 
 |-  ( s ` G ) = ( s ` ( iota_ g e. T ( g ` P ) = Q ) )
11 10 eqeq2i 
 |-  ( f = ( s ` G ) <-> f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) )
12 11 anbi1i 
 |-  ( ( f = ( s ` G ) /\ s e. E ) <-> ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) )
13 12 opabbii 
 |-  { <. f , s >. | ( f = ( s ` G ) /\ s e. E ) } = { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) }
14 9 13 eqtr4di 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) = { <. f , s >. | ( f = ( s ` G ) /\ s e. E ) } )