Step |
Hyp |
Ref |
Expression |
1 |
|
flimtop |
⊢ ( 𝑎 ∈ ( 𝐽 fLim 𝐹 ) → 𝐽 ∈ Top ) |
2 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
3 |
2
|
flimfil |
⊢ ( 𝑎 ∈ ( 𝐽 fLim 𝐹 ) → 𝐹 ∈ ( Fil ‘ ∪ 𝐽 ) ) |
4 |
|
flimclsi |
⊢ ( 𝑥 ∈ 𝐹 → ( 𝐽 fLim 𝐹 ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑥 ) ) |
5 |
4
|
sseld |
⊢ ( 𝑥 ∈ 𝐹 → ( 𝑎 ∈ ( 𝐽 fLim 𝐹 ) → 𝑎 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑥 ) ) ) |
6 |
5
|
com12 |
⊢ ( 𝑎 ∈ ( 𝐽 fLim 𝐹 ) → ( 𝑥 ∈ 𝐹 → 𝑎 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑥 ) ) ) |
7 |
6
|
ralrimiv |
⊢ ( 𝑎 ∈ ( 𝐽 fLim 𝐹 ) → ∀ 𝑥 ∈ 𝐹 𝑎 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑥 ) ) |
8 |
2
|
isfcls |
⊢ ( 𝑎 ∈ ( 𝐽 fClus 𝐹 ) ↔ ( 𝐽 ∈ Top ∧ 𝐹 ∈ ( Fil ‘ ∪ 𝐽 ) ∧ ∀ 𝑥 ∈ 𝐹 𝑎 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑥 ) ) ) |
9 |
1 3 7 8
|
syl3anbrc |
⊢ ( 𝑎 ∈ ( 𝐽 fLim 𝐹 ) → 𝑎 ∈ ( 𝐽 fClus 𝐹 ) ) |
10 |
9
|
ssriv |
⊢ ( 𝐽 fLim 𝐹 ) ⊆ ( 𝐽 fClus 𝐹 ) |