Theorem flfelbas

Description: A limit point of a function is in the topological space. (Contributed by Jeff Hankins, 10-Nov-2009) (Revised by Stefan O'Rear, 7-Aug-2015)

	Ref	Expression
Assertion	flfelbas	⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝐴 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ) → 𝐴 ∈ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		isflf	⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝐴 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ↔ ( 𝐴 ∈ 𝑋 ∧ ∀ 𝑜 ∈ 𝐽 ( 𝐴 ∈ 𝑜 → ∃ 𝑠 ∈ 𝐿 ( 𝐹 “ 𝑠 ) ⊆ 𝑜 ) ) ) )
2	1	simprbda	⊢ ( ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐿 ∈ ( Fil ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝐴 ∈ ( ( 𝐽 fLimf 𝐿 ) ‘ 𝐹 ) ) → 𝐴 ∈ 𝑋 )