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Metamath Proof Explorer
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 ) |