Theorem cldcls

Description: A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007)

	Ref	Expression
Assertion	cldcls	|- ( S e. ( Clsd ` J ) -> ( ( cls ` J ) ` S ) = S )

Proof

Step	Hyp	Ref	Expression
1		cldrcl	|- ( S e. ( Clsd ` J ) -> J e. Top )
2		eqid	|- U. J = U. J
3	2	cldss	|- ( S e. ( Clsd ` J ) -> S C_ U. J )
4	2	clsval	|- ( ( J e. Top /\ S C_ U. J ) -> ( ( cls ` J ) ` S ) = |^| { x e. ( Clsd ` J ) | S C_ x } )
5	1 3 4	syl2anc	|- ( S e. ( Clsd ` J ) -> ( ( cls ` J ) ` S ) = |^| { x e. ( Clsd ` J ) | S C_ x } )
6		intmin	|- ( S e. ( Clsd ` J ) -> |^| { x e. ( Clsd ` J ) | S C_ x } = S )
7	5 6	eqtrd	|- ( S e. ( Clsd ` J ) -> ( ( cls ` J ) ` S ) = S )