Metamath Proof Explorer


Theorem fclstop

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 ) ∧ ∀ 𝑠𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) )
3 2 simp1bi ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → 𝐽 ∈ Top )