Metamath Proof Explorer

Theorem lmcl

Description: Closure of a limit. (Contributed by NM, 19-Dec-2006) (Revised by Mario Carneiro, 23-Dec-2013)

Ref Expression
Assertion lmcl ⊢ J ∈ TopOn ⁡ X ∧ F ⇝t ⁡ J P → P ∈ X

Proof

Step Hyp Ref Expression
1 id ⊢ J ∈ TopOn ⁡ X → J ∈ TopOn ⁡ X
2 1 lmbr ⊢ J ∈ TopOn ⁡ X → F ⇝t ⁡ J P ↔ F ∈ X ↑ 𝑝𝑚 ℂ ∧ P ∈ X ∧ ∀ u ∈ J P ∈ u → ∃ y ∈ ran ⁡ ℤ ≥ F ↾ y : y ⟶ u
3 2 biimpa ⊢ J ∈ TopOn ⁡ X ∧ F ⇝t ⁡ J P → F ∈ X ↑ 𝑝𝑚 ℂ ∧ P ∈ X ∧ ∀ u ∈ J P ∈ u → ∃ y ∈ ran ⁡ ℤ ≥ F ↾ y : y ⟶ u
4 3 simp2d ⊢ J ∈ TopOn ⁡ X ∧ F ⇝t ⁡ J P → P ∈ X