Theorem flimneiss

Description: A filter contains the neighborhood filter as a subfilter. (Contributed by Mario Carneiro, 9-Apr-2015) (Revised by Stefan O'Rear, 9-Aug-2015)

	Ref	Expression
Assertion	flimneiss	|- ( A e. ( J fLim F ) -> ( ( nei ` J ) ` { A } ) C_ F )

Proof

Step	Hyp	Ref	Expression
1		eqid	|- U. J = U. J
2	1	elflim2	|- ( A e. ( J fLim F ) <-> ( ( J e. Top /\ F e. U. ran Fil /\ F C_ ~P U. J ) /\ ( A e. U. J /\ ( ( nei ` J ) ` { A } ) C_ F ) ) )
3	2	simprbi	|- ( A e. ( J fLim F ) -> ( A e. U. J /\ ( ( nei ` J ) ` { A } ) C_ F ) )
4	3	simprd	|- ( A e. ( J fLim F ) -> ( ( nei ` J ) ` { A } ) C_ F )