Step |
Hyp |
Ref |
Expression |
1 |
|
clscld.1 |
|- X = U. J |
2 |
|
difss |
|- ( X \ S ) C_ X |
3 |
1
|
clsval2 |
|- ( ( J e. Top /\ ( X \ S ) C_ X ) -> ( ( cls ` J ) ` ( X \ S ) ) = ( X \ ( ( int ` J ) ` ( X \ ( X \ S ) ) ) ) ) |
4 |
2 3
|
mpan2 |
|- ( J e. Top -> ( ( cls ` J ) ` ( X \ S ) ) = ( X \ ( ( int ` J ) ` ( X \ ( X \ S ) ) ) ) ) |
5 |
4
|
adantr |
|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` ( X \ S ) ) = ( X \ ( ( int ` J ) ` ( X \ ( X \ S ) ) ) ) ) |
6 |
|
dfss4 |
|- ( S C_ X <-> ( X \ ( X \ S ) ) = S ) |
7 |
6
|
biimpi |
|- ( S C_ X -> ( X \ ( X \ S ) ) = S ) |
8 |
7
|
fveq2d |
|- ( S C_ X -> ( ( int ` J ) ` ( X \ ( X \ S ) ) ) = ( ( int ` J ) ` S ) ) |
9 |
8
|
adantl |
|- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` ( X \ ( X \ S ) ) ) = ( ( int ` J ) ` S ) ) |
10 |
9
|
difeq2d |
|- ( ( J e. Top /\ S C_ X ) -> ( X \ ( ( int ` J ) ` ( X \ ( X \ S ) ) ) ) = ( X \ ( ( int ` J ) ` S ) ) ) |
11 |
5 10
|
eqtrd |
|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` ( X \ S ) ) = ( X \ ( ( int ` J ) ` S ) ) ) |
12 |
11
|
difeq2d |
|- ( ( J e. Top /\ S C_ X ) -> ( X \ ( ( cls ` J ) ` ( X \ S ) ) ) = ( X \ ( X \ ( ( int ` J ) ` S ) ) ) ) |
13 |
1
|
ntropn |
|- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) e. J ) |
14 |
1
|
eltopss |
|- ( ( J e. Top /\ ( ( int ` J ) ` S ) e. J ) -> ( ( int ` J ) ` S ) C_ X ) |
15 |
13 14
|
syldan |
|- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) C_ X ) |
16 |
|
dfss4 |
|- ( ( ( int ` J ) ` S ) C_ X <-> ( X \ ( X \ ( ( int ` J ) ` S ) ) ) = ( ( int ` J ) ` S ) ) |
17 |
15 16
|
sylib |
|- ( ( J e. Top /\ S C_ X ) -> ( X \ ( X \ ( ( int ` J ) ` S ) ) ) = ( ( int ` J ) ` S ) ) |
18 |
12 17
|
eqtr2d |
|- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = ( X \ ( ( cls ` J ) ` ( X \ S ) ) ) ) |