Step |
Hyp |
Ref |
Expression |
1 |
|
ntruni.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
|
elssuni |
⊢ ( 𝑜 ∈ 𝑂 → 𝑜 ⊆ ∪ 𝑂 ) |
3 |
|
sspwuni |
⊢ ( 𝑂 ⊆ 𝒫 𝑋 ↔ ∪ 𝑂 ⊆ 𝑋 ) |
4 |
1
|
ntrss |
⊢ ( ( 𝐽 ∈ Top ∧ ∪ 𝑂 ⊆ 𝑋 ∧ 𝑜 ⊆ ∪ 𝑂 ) → ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ ∪ 𝑂 ) ) |
5 |
4
|
3expia |
⊢ ( ( 𝐽 ∈ Top ∧ ∪ 𝑂 ⊆ 𝑋 ) → ( 𝑜 ⊆ ∪ 𝑂 → ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ ∪ 𝑂 ) ) ) |
6 |
3 5
|
sylan2b |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋 ) → ( 𝑜 ⊆ ∪ 𝑂 → ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ ∪ 𝑂 ) ) ) |
7 |
2 6
|
syl5 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋 ) → ( 𝑜 ∈ 𝑂 → ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ ∪ 𝑂 ) ) ) |
8 |
7
|
ralrimiv |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋 ) → ∀ 𝑜 ∈ 𝑂 ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ ∪ 𝑂 ) ) |
9 |
|
iunss |
⊢ ( ∪ 𝑜 ∈ 𝑂 ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ ∪ 𝑂 ) ↔ ∀ 𝑜 ∈ 𝑂 ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ ∪ 𝑂 ) ) |
10 |
8 9
|
sylibr |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑂 ⊆ 𝒫 𝑋 ) → ∪ 𝑜 ∈ 𝑂 ( ( int ‘ 𝐽 ) ‘ 𝑜 ) ⊆ ( ( int ‘ 𝐽 ) ‘ ∪ 𝑂 ) ) |