Step |
Hyp |
Ref |
Expression |
1 |
|
restcls2.1 |
⊢ ( 𝜑 → 𝐽 ∈ Top ) |
2 |
|
restcls2.2 |
⊢ ( 𝜑 → 𝑋 = ∪ 𝐽 ) |
3 |
|
restcls2.3 |
⊢ ( 𝜑 → 𝑌 ⊆ 𝑋 ) |
4 |
|
restcls2.4 |
⊢ ( 𝜑 → 𝐾 = ( 𝐽 ↾t 𝑌 ) ) |
5 |
|
restcls2.5 |
⊢ ( 𝜑 → 𝑆 ∈ ( Clsd ‘ 𝐾 ) ) |
6 |
|
restclsseplem.6 |
⊢ ( 𝜑 → ( 𝑆 ∩ 𝑇 ) = ∅ ) |
7 |
|
restclsseplem.7 |
⊢ ( 𝜑 → 𝑇 ⊆ 𝑌 ) |
8 |
1 2 3 4 5
|
restcls2 |
⊢ ( 𝜑 → 𝑆 = ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑌 ) ) |
9 |
8
|
ineq1d |
⊢ ( 𝜑 → ( 𝑆 ∩ 𝑇 ) = ( ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑌 ) ∩ 𝑇 ) ) |
10 |
|
inass |
⊢ ( ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑌 ) ∩ 𝑇 ) = ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( 𝑌 ∩ 𝑇 ) ) |
11 |
9 10
|
eqtrdi |
⊢ ( 𝜑 → ( 𝑆 ∩ 𝑇 ) = ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( 𝑌 ∩ 𝑇 ) ) ) |
12 |
|
sseqin2 |
⊢ ( 𝑇 ⊆ 𝑌 ↔ ( 𝑌 ∩ 𝑇 ) = 𝑇 ) |
13 |
7 12
|
sylib |
⊢ ( 𝜑 → ( 𝑌 ∩ 𝑇 ) = 𝑇 ) |
14 |
13
|
ineq2d |
⊢ ( 𝜑 → ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ ( 𝑌 ∩ 𝑇 ) ) = ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑇 ) ) |
15 |
11 6 14
|
3eqtr3rd |
⊢ ( 𝜑 → ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑇 ) = ∅ ) |