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Metamath Proof Explorer

Theorem dicopelval

Description: Membership in value of the partial isomorphism C for a lattice K . (Contributed by NM, 15-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 )
dicelval.f 
|- F e. _V
dicelval.s 
|- S e. _V
Assertion dicopelval 
|- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. F , S >. e. ( I ` Q ) <-> ( F = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ 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 dicelval.f 
 |-  F e. _V
9 dicelval.s 
 |-  S e. _V
10 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 ) } )
11 10 eleq2d 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. F , S >. e. ( I ` Q ) <-> <. F , S >. e. { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) } ) )
12 eqeq1 
 |-  ( f = F -> ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) <-> F = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) ) )
13 12 anbi1d 
 |-  ( f = F -> ( ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) <-> ( F = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) ) )
14 fveq1 
 |-  ( s = S -> ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) )
15 14 eqeq2d 
 |-  ( s = S -> ( F = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) <-> F = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) ) )
16 eleq1 
 |-  ( s = S -> ( s e. E <-> S e. E ) )
17 15 16 anbi12d 
 |-  ( s = S -> ( ( F = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) <-> ( F = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ S e. E ) ) )
18 8 9 13 17 opelopab 
 |-  ( <. F , S >. e. { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) } <-> ( F = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ S e. E ) )
19 11 18 bitrdi 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. F , S >. e. ( I ` Q ) <-> ( F = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ S e. E ) ) )