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 |
4
|
fveq2d |
⊢ ( 𝜑 → ( cls ‘ 𝐾 ) = ( cls ‘ ( 𝐽 ↾t 𝑌 ) ) ) |
7 |
6
|
fveq1d |
⊢ ( 𝜑 → ( ( cls ‘ 𝐾 ) ‘ 𝑆 ) = ( ( cls ‘ ( 𝐽 ↾t 𝑌 ) ) ‘ 𝑆 ) ) |
8 |
|
cldcls |
⊢ ( 𝑆 ∈ ( Clsd ‘ 𝐾 ) → ( ( cls ‘ 𝐾 ) ‘ 𝑆 ) = 𝑆 ) |
9 |
5 8
|
syl |
⊢ ( 𝜑 → ( ( cls ‘ 𝐾 ) ‘ 𝑆 ) = 𝑆 ) |
10 |
3 2
|
sseqtrd |
⊢ ( 𝜑 → 𝑌 ⊆ ∪ 𝐽 ) |
11 |
1 2 3 4 5
|
restcls2lem |
⊢ ( 𝜑 → 𝑆 ⊆ 𝑌 ) |
12 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
13 |
|
eqid |
⊢ ( 𝐽 ↾t 𝑌 ) = ( 𝐽 ↾t 𝑌 ) |
14 |
12 13
|
restcls |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑌 ⊆ ∪ 𝐽 ∧ 𝑆 ⊆ 𝑌 ) → ( ( cls ‘ ( 𝐽 ↾t 𝑌 ) ) ‘ 𝑆 ) = ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑌 ) ) |
15 |
1 10 11 14
|
syl3anc |
⊢ ( 𝜑 → ( ( cls ‘ ( 𝐽 ↾t 𝑌 ) ) ‘ 𝑆 ) = ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑌 ) ) |
16 |
7 9 15
|
3eqtr3d |
⊢ ( 𝜑 → 𝑆 = ( ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ∩ 𝑌 ) ) |