Step |
Hyp |
Ref |
Expression |
1 |
|
clscld.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
|
difss |
⊢ ( 𝑋 ∖ 𝑆 ) ⊆ 𝑋 |
3 |
1
|
clsval2 |
⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑋 ∖ 𝑆 ) ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ ( 𝑋 ∖ 𝑆 ) ) = ( 𝑋 ∖ ( ( int ‘ 𝐽 ) ‘ ( 𝑋 ∖ ( 𝑋 ∖ 𝑆 ) ) ) ) ) |
4 |
2 3
|
mpan2 |
⊢ ( 𝐽 ∈ Top → ( ( cls ‘ 𝐽 ) ‘ ( 𝑋 ∖ 𝑆 ) ) = ( 𝑋 ∖ ( ( int ‘ 𝐽 ) ‘ ( 𝑋 ∖ ( 𝑋 ∖ 𝑆 ) ) ) ) ) |
5 |
4
|
adantr |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ ( 𝑋 ∖ 𝑆 ) ) = ( 𝑋 ∖ ( ( int ‘ 𝐽 ) ‘ ( 𝑋 ∖ ( 𝑋 ∖ 𝑆 ) ) ) ) ) |
6 |
|
dfss4 |
⊢ ( 𝑆 ⊆ 𝑋 ↔ ( 𝑋 ∖ ( 𝑋 ∖ 𝑆 ) ) = 𝑆 ) |
7 |
6
|
biimpi |
⊢ ( 𝑆 ⊆ 𝑋 → ( 𝑋 ∖ ( 𝑋 ∖ 𝑆 ) ) = 𝑆 ) |
8 |
7
|
fveq2d |
⊢ ( 𝑆 ⊆ 𝑋 → ( ( int ‘ 𝐽 ) ‘ ( 𝑋 ∖ ( 𝑋 ∖ 𝑆 ) ) ) = ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ) |
9 |
8
|
adantl |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ ( 𝑋 ∖ ( 𝑋 ∖ 𝑆 ) ) ) = ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ) |
10 |
9
|
difeq2d |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑋 ∖ ( ( int ‘ 𝐽 ) ‘ ( 𝑋 ∖ ( 𝑋 ∖ 𝑆 ) ) ) ) = ( 𝑋 ∖ ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ) ) |
11 |
5 10
|
eqtrd |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ ( 𝑋 ∖ 𝑆 ) ) = ( 𝑋 ∖ ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ) ) |
12 |
11
|
difeq2d |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ ( 𝑋 ∖ 𝑆 ) ) ) = ( 𝑋 ∖ ( 𝑋 ∖ ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ) ) ) |
13 |
1
|
ntropn |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ∈ 𝐽 ) |
14 |
1
|
eltopss |
⊢ ( ( 𝐽 ∈ Top ∧ ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ∈ 𝐽 ) → ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ⊆ 𝑋 ) |
15 |
13 14
|
syldan |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ⊆ 𝑋 ) |
16 |
|
dfss4 |
⊢ ( ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ⊆ 𝑋 ↔ ( 𝑋 ∖ ( 𝑋 ∖ ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ) ) = ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ) |
17 |
15 16
|
sylib |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( 𝑋 ∖ ( 𝑋 ∖ ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ) ) = ( ( int ‘ 𝐽 ) ‘ 𝑆 ) ) |
18 |
12 17
|
eqtr2d |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( int ‘ 𝐽 ) ‘ 𝑆 ) = ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ ( 𝑋 ∖ 𝑆 ) ) ) ) |