| Step |
Hyp |
Ref |
Expression |
| 1 |
|
patoms.b |
|- B = ( Base ` K ) |
| 2 |
|
patoms.z |
|- .0. = ( 0. ` K ) |
| 3 |
|
patoms.c |
|- C = ( |
| 4 |
|
patoms.a |
|- A = ( Atoms ` K ) |
| 5 |
|
elex |
|- ( K e. D -> K e. _V ) |
| 6 |
|
fveq2 |
|- ( p = K -> ( Base ` p ) = ( Base ` K ) ) |
| 7 |
6 1
|
eqtr4di |
|- ( p = K -> ( Base ` p ) = B ) |
| 8 |
|
fveq2 |
|- ( p = K -> ( |
| 9 |
8 3
|
eqtr4di |
|- ( p = K -> ( |
| 10 |
9
|
breqd |
|- ( p = K -> ( ( 0. ` p ) ( ( 0. ` p ) C x ) ) |
| 11 |
|
fveq2 |
|- ( p = K -> ( 0. ` p ) = ( 0. ` K ) ) |
| 12 |
11 2
|
eqtr4di |
|- ( p = K -> ( 0. ` p ) = .0. ) |
| 13 |
12
|
breq1d |
|- ( p = K -> ( ( 0. ` p ) C x <-> .0. C x ) ) |
| 14 |
10 13
|
bitrd |
|- ( p = K -> ( ( 0. ` p ) ( .0. C x ) ) |
| 15 |
7 14
|
rabeqbidv |
|- ( p = K -> { x e. ( Base ` p ) | ( 0. ` p ) ( |
| 16 |
|
df-ats |
|- Atoms = ( p e. _V |-> { x e. ( Base ` p ) | ( 0. ` p ) ( |
| 17 |
1
|
fvexi |
|- B e. _V |
| 18 |
17
|
rabex |
|- { x e. B | .0. C x } e. _V |
| 19 |
15 16 18
|
fvmpt |
|- ( K e. _V -> ( Atoms ` K ) = { x e. B | .0. C x } ) |
| 20 |
4 19
|
eqtrid |
|- ( K e. _V -> A = { x e. B | .0. C x } ) |
| 21 |
5 20
|
syl |
|- ( K e. D -> A = { x e. B | .0. C x } ) |