Metamath Proof Explorer


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 )