Step |
Hyp |
Ref |
Expression |
1 |
|
watomfval.a |
|- A = ( Atoms ` K ) |
2 |
|
watomfval.p |
|- P = ( _|_P ` K ) |
3 |
|
watomfval.w |
|- W = ( WAtoms ` K ) |
4 |
1 2 3
|
watfvalN |
|- ( K e. B -> W = ( d e. A |-> ( A \ ( ( _|_P ` K ) ` { d } ) ) ) ) |
5 |
4
|
fveq1d |
|- ( K e. B -> ( W ` D ) = ( ( d e. A |-> ( A \ ( ( _|_P ` K ) ` { d } ) ) ) ` D ) ) |
6 |
|
sneq |
|- ( d = D -> { d } = { D } ) |
7 |
6
|
fveq2d |
|- ( d = D -> ( ( _|_P ` K ) ` { d } ) = ( ( _|_P ` K ) ` { D } ) ) |
8 |
7
|
difeq2d |
|- ( d = D -> ( A \ ( ( _|_P ` K ) ` { d } ) ) = ( A \ ( ( _|_P ` K ) ` { D } ) ) ) |
9 |
|
eqid |
|- ( d e. A |-> ( A \ ( ( _|_P ` K ) ` { d } ) ) ) = ( d e. A |-> ( A \ ( ( _|_P ` K ) ` { d } ) ) ) |
10 |
1
|
fvexi |
|- A e. _V |
11 |
10
|
difexi |
|- ( A \ ( ( _|_P ` K ) ` { D } ) ) e. _V |
12 |
8 9 11
|
fvmpt |
|- ( D e. A -> ( ( d e. A |-> ( A \ ( ( _|_P ` K ) ` { d } ) ) ) ` D ) = ( A \ ( ( _|_P ` K ) ` { D } ) ) ) |
13 |
5 12
|
sylan9eq |
|- ( ( K e. B /\ D e. A ) -> ( W ` D ) = ( A \ ( ( _|_P ` K ) ` { D } ) ) ) |