This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: Reverse closure for the cluster point predicate. (Contributed by Mario
Carneiro, 11-Apr-2015) (Revised by Stefan O'Rear, 8-Aug-2015)
|
|
Ref |
Expression |
|
Assertion |
fclstop |
⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → 𝐽 ∈ Top ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
| 2 |
1
|
isfcls |
⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) ↔ ( 𝐽 ∈ Top ∧ 𝐹 ∈ ( Fil ‘ ∪ 𝐽 ) ∧ ∀ 𝑠 ∈ 𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) ) |
| 3 |
2
|
simp1bi |
⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → 𝐽 ∈ Top ) |