Theorem fclsfil

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
Hypothesis	fclsval.x	⊢ 𝑋 = ∪ 𝐽
Assertion	fclsfil	⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		fclsval.x	⊢ 𝑋 = ∪ 𝐽
2	1	isfcls	⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) ↔ ( 𝐽 ∈ Top ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝐹 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ) )
3	2	simp2bi	⊢ ( 𝐴 ∈ ( 𝐽 fClus 𝐹 ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) )