Theorem ntrtop

Description: The interior of a topology's underlying set is the entire set. (Contributed by NM, 12-Sep-2006)

	Ref	Expression
Hypothesis	clscld.1	|- X = U. J
Assertion	ntrtop	|- ( J e. Top -> ( ( int ` J ) ` X ) = X )

Proof

Step	Hyp	Ref	Expression
1		clscld.1	|- X = U. J
2	1	topopn	|- ( J e. Top -> X e. J )
3		ssid	|- X C_ X
4	1	isopn3	|- ( ( J e. Top /\ X C_ X ) -> ( X e. J <-> ( ( int ` J ) ` X ) = X ) )
5	3 4	mpan2	|- ( J e. Top -> ( X e. J <-> ( ( int ` J ) ` X ) = X ) )
6	2 5	mpbid	|- ( J e. Top -> ( ( int ` J ) ` X ) = X )