Metamath Proof Explorer

Theorem dicelvalN

Description: Membership in value of the partial isomorphism C for a lattice K . (Contributed by NM, 25-Feb-2014) (New usage is discouraged.)

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 )
Assertion dicelvalN 
|- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Y e. ( I ` Q ) <-> ( Y e. ( _V X. _V ) /\ ( ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ ( 2nd ` Y ) 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 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 ) } )
9 8 eleq2d 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Y e. ( I ` Q ) <-> Y e. { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) } ) )
10 vex 
 |-  f e. _V
11 vex 
 |-  s e. _V
12 10 11 op1std 
 |-  ( Y = <. f , s >. -> ( 1st ` Y ) = f )
13 10 11 op2ndd 
 |-  ( Y = <. f , s >. -> ( 2nd ` Y ) = s )
14 13 fveq1d 
 |-  ( Y = <. f , s >. -> ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) ) = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) )
15 12 14 eqeq12d 
 |-  ( Y = <. f , s >. -> ( ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) ) <-> f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) ) )
16 13 eleq1d 
 |-  ( Y = <. f , s >. -> ( ( 2nd ` Y ) e. E <-> s e. E ) )
17 15 16 anbi12d 
 |-  ( Y = <. f , s >. -> ( ( ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ ( 2nd ` Y ) e. E ) <-> ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) ) )
18 17 elopaba 
 |-  ( Y e. { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) } <-> ( Y e. ( _V X. _V ) /\ ( ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ ( 2nd ` Y ) e. E ) ) )
19 9 18 bitrdi 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Y e. ( I ` Q ) <-> ( Y e. ( _V X. _V ) /\ ( ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ ( 2nd ` Y ) e. E ) ) ) )