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Metamath Proof Explorer

Theorem lmfpm

Description: If F converges, then F is a partial function. (Contributed by Mario Carneiro, 23-Dec-2013)

Ref Expression
Assertion lmfpm ⊢ J ∈ TopOn ⁡ X ∧ F ⇝t ⁡ J P → F ∈ X ↑ 𝑝𝑚 ℂ

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 simp1d ⊢ J ∈ TopOn ⁡ X ∧ F ⇝t ⁡ J P → F ∈ X ↑ 𝑝𝑚 ℂ