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Metamath Proof Explorer

Theorem dicelval2N

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 )
dicval2.g 
|- G = ( iota_ g e. T ( g ` P ) = Q )
Assertion dicelval2N 
|- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Y e. ( I ` Q ) <-> ( Y e. ( _V X. _V ) /\ ( ( 1st ` Y ) = ( ( 2nd ` Y ) ` G ) /\ ( 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 dicval2.g 
 |-  G = ( iota_ g e. T ( g ` P ) = Q )
9 1 2 3 4 5 6 7 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 ) ) ) )
10 8 fveq2i 
 |-  ( ( 2nd ` Y ) ` G ) = ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) )
11 10 eqeq2i 
 |-  ( ( 1st ` Y ) = ( ( 2nd ` Y ) ` G ) <-> ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) ) )
12 11 anbi1i 
 |-  ( ( ( 1st ` Y ) = ( ( 2nd ` Y ) ` G ) /\ ( 2nd ` Y ) e. E ) <-> ( ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ ( 2nd ` Y ) e. E ) )
13 12 anbi2i 
 |-  ( ( Y e. ( _V X. _V ) /\ ( ( 1st ` Y ) = ( ( 2nd ` Y ) ` G ) /\ ( 2nd ` Y ) e. E ) ) <-> ( Y e. ( _V X. _V ) /\ ( ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ ( 2nd ` Y ) e. E ) ) )
14 9 13 bitr4di 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Y e. ( I ` Q ) <-> ( Y e. ( _V X. _V ) /\ ( ( 1st ` Y ) = ( ( 2nd ` Y ) ` G ) /\ ( 2nd ` Y ) e. E ) ) ) )