Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
2 |
1
|
isfcls |
⊢ ( 𝑥 ∈ ( 𝐽 fClus 𝐹 ) ↔ ( 𝐽 ∈ Top ∧ 𝐹 ∈ ( Fil ‘ ∪ 𝐽 ) ∧ ∀ 𝑠 ∈ 𝐹 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) ) |
3 |
2
|
simp3bi |
⊢ ( 𝑥 ∈ ( 𝐽 fClus 𝐹 ) → ∀ 𝑠 ∈ 𝐹 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) |
4 |
|
fveq2 |
⊢ ( 𝑠 = 𝑆 → ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) = ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) |
5 |
4
|
eleq2d |
⊢ ( 𝑠 = 𝑆 → ( 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ↔ 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ) |
6 |
5
|
rspcv |
⊢ ( 𝑆 ∈ 𝐹 → ( ∀ 𝑠 ∈ 𝐹 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) → 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ) |
7 |
3 6
|
syl5 |
⊢ ( 𝑆 ∈ 𝐹 → ( 𝑥 ∈ ( 𝐽 fClus 𝐹 ) → 𝑥 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ) |
8 |
7
|
ssrdv |
⊢ ( 𝑆 ∈ 𝐹 → ( 𝐽 fClus 𝐹 ) ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) |