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 |
|
dicelval2.f |
|- F e. _V |
10 |
|
dicelval2.s |
|- S e. _V |
11 |
1 2 3 4 5 6 7 9 10
|
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 ) ) ) |
12 |
8
|
fveq2i |
|- ( S ` G ) = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) |
13 |
12
|
eqeq2i |
|- ( F = ( S ` G ) <-> F = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) ) |
14 |
13
|
anbi1i |
|- ( ( F = ( S ` G ) /\ S e. E ) <-> ( F = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ S e. E ) ) |
15 |
11 14
|
bitr4di |
|- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. F , S >. e. ( I ` Q ) <-> ( F = ( S ` G ) /\ S e. E ) ) ) |