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 e. ( J fClus F ) -> J e. Top )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  U. J = U. J
2 1 isfcls 
 |-  ( A e. ( J fClus F ) <-> ( J e. Top /\ F e. ( Fil ` U. J ) /\ A. s e. F A e. ( ( cls ` J ) ` s ) ) )
3 2 simp1bi 
 |-  ( A e. ( J fClus F ) -> J e. Top )