Theorem ntridm

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

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

Proof

Step	Hyp	Ref	Expression
1		clscld.1	|- X = U. J
2	1	ntropn	|- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) e. J )
3	1	ntrss3	|- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) C_ X )
4	1	isopn3	|- ( ( J e. Top /\ ( ( int ` J ) ` S ) C_ X ) -> ( ( ( int ` J ) ` S ) e. J <-> ( ( int ` J ) ` ( ( int ` J ) ` S ) ) = ( ( int ` J ) ` S ) ) )
5	3 4	syldan	|- ( ( J e. Top /\ S C_ X ) -> ( ( ( int ` J ) ` S ) e. J <-> ( ( int ` J ) ` ( ( int ` J ) ` S ) ) = ( ( int ` J ) ` S ) ) )
6	2 5	mpbid	|- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` ( ( int ` J ) ` S ) ) = ( ( int ` J ) ` S ) )