Step |
Hyp |
Ref |
Expression |
1 |
|
fvex |
|- ( Clsd ` J ) e. _V |
2 |
1
|
elpw2 |
|- ( X e. ~P ( Clsd ` J ) <-> X C_ ( Clsd ` J ) ) |
3 |
2
|
biimpri |
|- ( X C_ ( Clsd ` J ) -> X e. ~P ( Clsd ` J ) ) |
4 |
|
cmptop |
|- ( J e. Comp -> J e. Top ) |
5 |
|
cmpfi |
|- ( J e. Top -> ( J e. Comp <-> A. x e. ~P ( Clsd ` J ) ( -. (/) e. ( fi ` x ) -> |^| x =/= (/) ) ) ) |
6 |
4 5
|
syl |
|- ( J e. Comp -> ( J e. Comp <-> A. x e. ~P ( Clsd ` J ) ( -. (/) e. ( fi ` x ) -> |^| x =/= (/) ) ) ) |
7 |
6
|
ibi |
|- ( J e. Comp -> A. x e. ~P ( Clsd ` J ) ( -. (/) e. ( fi ` x ) -> |^| x =/= (/) ) ) |
8 |
|
fveq2 |
|- ( x = X -> ( fi ` x ) = ( fi ` X ) ) |
9 |
8
|
eleq2d |
|- ( x = X -> ( (/) e. ( fi ` x ) <-> (/) e. ( fi ` X ) ) ) |
10 |
9
|
notbid |
|- ( x = X -> ( -. (/) e. ( fi ` x ) <-> -. (/) e. ( fi ` X ) ) ) |
11 |
|
inteq |
|- ( x = X -> |^| x = |^| X ) |
12 |
11
|
neeq1d |
|- ( x = X -> ( |^| x =/= (/) <-> |^| X =/= (/) ) ) |
13 |
10 12
|
imbi12d |
|- ( x = X -> ( ( -. (/) e. ( fi ` x ) -> |^| x =/= (/) ) <-> ( -. (/) e. ( fi ` X ) -> |^| X =/= (/) ) ) ) |
14 |
13
|
rspcva |
|- ( ( X e. ~P ( Clsd ` J ) /\ A. x e. ~P ( Clsd ` J ) ( -. (/) e. ( fi ` x ) -> |^| x =/= (/) ) ) -> ( -. (/) e. ( fi ` X ) -> |^| X =/= (/) ) ) |
15 |
3 7 14
|
syl2anr |
|- ( ( J e. Comp /\ X C_ ( Clsd ` J ) ) -> ( -. (/) e. ( fi ` X ) -> |^| X =/= (/) ) ) |
16 |
15
|
3impia |
|- ( ( J e. Comp /\ X C_ ( Clsd ` J ) /\ -. (/) e. ( fi ` X ) ) -> |^| X =/= (/) ) |