Theorem flimelbas

Description: A limit point of a filter belongs to its base set. (Contributed by Jeff Hankins, 4-Sep-2009) (Revised by Mario Carneiro, 9-Apr-2015)

	Ref	Expression
Hypothesis	flimuni.1	|- X = U. J
Assertion	flimelbas	|- ( A e. ( J fLim F ) -> A e. X )

Proof

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