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 = U. J
Assertion	fclsfil	|- ( A e. ( J fClus F ) -> F e. ( Fil ` X ) )

Proof

Step	Hyp	Ref	Expression
1		fclsval.x	|- X = U. J
2	1	isfcls	|- ( A e. ( J fClus F ) <-> ( J e. Top /\ F e. ( Fil ` X ) /\ A. s e. F A e. ( ( cls ` J ) ` s ) ) )
3	2	simp2bi	|- ( A e. ( J fClus F ) -> F e. ( Fil ` X ) )