Step |
Hyp |
Ref |
Expression |
1 |
|
lpfval.1 |
|- X = U. J |
2 |
|
simp1 |
|- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> J e. Top ) |
3 |
|
simp2 |
|- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> S C_ X ) |
4 |
3
|
ssdifssd |
|- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( S \ { x } ) C_ X ) |
5 |
|
simp3 |
|- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> T C_ S ) |
6 |
5
|
ssdifd |
|- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( T \ { x } ) C_ ( S \ { x } ) ) |
7 |
1
|
clsss |
|- ( ( J e. Top /\ ( S \ { x } ) C_ X /\ ( T \ { x } ) C_ ( S \ { x } ) ) -> ( ( cls ` J ) ` ( T \ { x } ) ) C_ ( ( cls ` J ) ` ( S \ { x } ) ) ) |
8 |
2 4 6 7
|
syl3anc |
|- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( cls ` J ) ` ( T \ { x } ) ) C_ ( ( cls ` J ) ` ( S \ { x } ) ) ) |
9 |
8
|
sseld |
|- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( x e. ( ( cls ` J ) ` ( T \ { x } ) ) -> x e. ( ( cls ` J ) ` ( S \ { x } ) ) ) ) |
10 |
5 3
|
sstrd |
|- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> T C_ X ) |
11 |
1
|
islp |
|- ( ( J e. Top /\ T C_ X ) -> ( x e. ( ( limPt ` J ) ` T ) <-> x e. ( ( cls ` J ) ` ( T \ { x } ) ) ) ) |
12 |
2 10 11
|
syl2anc |
|- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( x e. ( ( limPt ` J ) ` T ) <-> x e. ( ( cls ` J ) ` ( T \ { x } ) ) ) ) |
13 |
1
|
islp |
|- ( ( J e. Top /\ S C_ X ) -> ( x e. ( ( limPt ` J ) ` S ) <-> x e. ( ( cls ` J ) ` ( S \ { x } ) ) ) ) |
14 |
2 3 13
|
syl2anc |
|- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( x e. ( ( limPt ` J ) ` S ) <-> x e. ( ( cls ` J ) ` ( S \ { x } ) ) ) ) |
15 |
9 12 14
|
3imtr4d |
|- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( x e. ( ( limPt ` J ) ` T ) -> x e. ( ( limPt ` J ) ` S ) ) ) |
16 |
15
|
ssrdv |
|- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( limPt ` J ) ` T ) C_ ( ( limPt ` J ) ` S ) ) |