Theorem clsidm

Description: The closure operation is idempotent. (Contributed by NM, 2-Oct-2007)

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

Proof

Step	Hyp	Ref	Expression
1		clscld.1	|- X = U. J
2	1	clscld	|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) e. ( Clsd ` J ) )
3	1	clsss3	|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) C_ X )
4	1	iscld3	|- ( ( J e. Top /\ ( ( cls ` J ) ` S ) C_ X ) -> ( ( ( cls ` J ) ` S ) e. ( Clsd ` J ) <-> ( ( cls ` J ) ` ( ( cls ` J ) ` S ) ) = ( ( cls ` J ) ` S ) ) )
5	3 4	syldan	|- ( ( J e. Top /\ S C_ X ) -> ( ( ( cls ` J ) ` S ) e. ( Clsd ` J ) <-> ( ( cls ` J ) ` ( ( cls ` J ) ` S ) ) = ( ( cls ` J ) ` S ) ) )
6	2 5	mpbid	|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` ( ( cls ` J ) ` S ) ) = ( ( cls ` J ) ` S ) )