Step |
Hyp |
Ref |
Expression |
1 |
|
watomfval.a |
|- A = ( Atoms ` K ) |
2 |
|
watomfval.p |
|- P = ( _|_P ` K ) |
3 |
|
watomfval.w |
|- W = ( WAtoms ` K ) |
4 |
|
elex |
|- ( K e. B -> K e. _V ) |
5 |
|
fveq2 |
|- ( k = K -> ( Atoms ` k ) = ( Atoms ` K ) ) |
6 |
5 1
|
eqtr4di |
|- ( k = K -> ( Atoms ` k ) = A ) |
7 |
|
fveq2 |
|- ( k = K -> ( _|_P ` k ) = ( _|_P ` K ) ) |
8 |
7
|
fveq1d |
|- ( k = K -> ( ( _|_P ` k ) ` { d } ) = ( ( _|_P ` K ) ` { d } ) ) |
9 |
6 8
|
difeq12d |
|- ( k = K -> ( ( Atoms ` k ) \ ( ( _|_P ` k ) ` { d } ) ) = ( A \ ( ( _|_P ` K ) ` { d } ) ) ) |
10 |
6 9
|
mpteq12dv |
|- ( k = K -> ( d e. ( Atoms ` k ) |-> ( ( Atoms ` k ) \ ( ( _|_P ` k ) ` { d } ) ) ) = ( d e. A |-> ( A \ ( ( _|_P ` K ) ` { d } ) ) ) ) |
11 |
|
df-watsN |
|- WAtoms = ( k e. _V |-> ( d e. ( Atoms ` k ) |-> ( ( Atoms ` k ) \ ( ( _|_P ` k ) ` { d } ) ) ) ) |
12 |
10 11 1
|
mptfvmpt |
|- ( K e. _V -> ( WAtoms ` K ) = ( d e. A |-> ( A \ ( ( _|_P ` K ) ` { d } ) ) ) ) |
13 |
3 12
|
syl5eq |
|- ( K e. _V -> W = ( d e. A |-> ( A \ ( ( _|_P ` K ) ` { d } ) ) ) ) |
14 |
4 13
|
syl |
|- ( K e. B -> W = ( d e. A |-> ( A \ ( ( _|_P ` K ) ` { d } ) ) ) ) |