Theorem iscld4

Description: A subset is closed iff it contains its own closure. (Contributed by NM, 31-Jan-2008)

	Ref	Expression
Hypothesis	clscld.1	|- X = U. J
Assertion	iscld4	|- ( ( J e. Top /\ S C_ X ) -> ( S e. ( Clsd ` J ) <-> ( ( cls ` J ) ` S ) C_ S ) )

Proof

Step	Hyp	Ref	Expression
1		clscld.1	|- X = U. J
2	1	iscld3	|- ( ( J e. Top /\ S C_ X ) -> ( S e. ( Clsd ` J ) <-> ( ( cls ` J ) ` S ) = S ) )
3		eqss	|- ( ( ( cls ` J ) ` S ) = S <-> ( ( ( cls ` J ) ` S ) C_ S /\ S C_ ( ( cls ` J ) ` S ) ) )
4	1	sscls	|- ( ( J e. Top /\ S C_ X ) -> S C_ ( ( cls ` J ) ` S ) )
5	4	biantrud	|- ( ( J e. Top /\ S C_ X ) -> ( ( ( cls ` J ) ` S ) C_ S <-> ( ( ( cls ` J ) ` S ) C_ S /\ S C_ ( ( cls ` J ) ` S ) ) ) )
6	3 5	bitr4id	|- ( ( J e. Top /\ S C_ X ) -> ( ( ( cls ` J ) ` S ) = S <-> ( ( cls ` J ) ` S ) C_ S ) )
7	2 6	bitrd	|- ( ( J e. Top /\ S C_ X ) -> ( S e. ( Clsd ` J ) <-> ( ( cls ` J ) ` S ) C_ S ) )