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 ∈ ( 𝐽 fLim 𝐹 ) → ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ⊆ 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ∪ 𝐽 = ∪ 𝐽
2	1	elflim2	⊢ ( 𝐴 ∈ ( 𝐽 fLim 𝐹 ) ↔ ( ( 𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 ∪ 𝐽 ) ∧ ( 𝐴 ∈ ∪ 𝐽 ∧ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ⊆ 𝐹 ) ) )
3	2	simprbi	⊢ ( 𝐴 ∈ ( 𝐽 fLim 𝐹 ) → ( 𝐴 ∈ ∪ 𝐽 ∧ ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ⊆ 𝐹 ) )
4	3	simprd	⊢ ( 𝐴 ∈ ( 𝐽 fLim 𝐹 ) → ( ( nei ‘ 𝐽 ) ‘ { 𝐴 } ) ⊆ 𝐹 )