| Step |
Hyp |
Ref |
Expression |
| 1 |
|
llnset.b |
|- B = ( Base ` K ) |
| 2 |
|
llnset.c |
|- C = ( |
| 3 |
|
llnset.a |
|- A = ( Atoms ` K ) |
| 4 |
|
llnset.n |
|- N = ( LLines ` K ) |
| 5 |
|
elex |
|- ( K e. D -> K e. _V ) |
| 6 |
|
fveq2 |
|- ( k = K -> ( Base ` k ) = ( Base ` K ) ) |
| 7 |
6 1
|
eqtr4di |
|- ( k = K -> ( Base ` k ) = B ) |
| 8 |
|
fveq2 |
|- ( k = K -> ( Atoms ` k ) = ( Atoms ` K ) ) |
| 9 |
8 3
|
eqtr4di |
|- ( k = K -> ( Atoms ` k ) = A ) |
| 10 |
|
fveq2 |
|- ( k = K -> ( |
| 11 |
10 2
|
eqtr4di |
|- ( k = K -> ( |
| 12 |
11
|
breqd |
|- ( k = K -> ( p ( p C x ) ) |
| 13 |
9 12
|
rexeqbidv |
|- ( k = K -> ( E. p e. ( Atoms ` k ) p ( E. p e. A p C x ) ) |
| 14 |
7 13
|
rabeqbidv |
|- ( k = K -> { x e. ( Base ` k ) | E. p e. ( Atoms ` k ) p ( |
| 15 |
|
df-llines |
|- LLines = ( k e. _V |-> { x e. ( Base ` k ) | E. p e. ( Atoms ` k ) p ( |
| 16 |
1
|
fvexi |
|- B e. _V |
| 17 |
16
|
rabex |
|- { x e. B | E. p e. A p C x } e. _V |
| 18 |
14 15 17
|
fvmpt |
|- ( K e. _V -> ( LLines ` K ) = { x e. B | E. p e. A p C x } ) |
| 19 |
4 18
|
eqtrid |
|- ( K e. _V -> N = { x e. B | E. p e. A p C x } ) |
| 20 |
5 19
|
syl |
|- ( K e. D -> N = { x e. B | E. p e. A p C x } ) |