have : (pure x : Filter α) ≤ φ (f x) := le_principal_iff.mpr ⟨x.2, le_refl (f x)⟩
case refine'_2
α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
f : α → β
inst✝¹ : LinearOrder β
inst✝ : OrderClosedTopology β
s : Set α
ne_s : Set.Nonempty s
hs : IsCompact s
hf : LowerSemicontinuousOn f s
this✝ : Nonempty α
this : Nonempty ↑s
φ : β → Filter α := fun b => 𝓟 (s ∩ f ⁻¹' Iic b)
ℱ : Filter α := ⨅ a, φ (f ↑a)
x : ↑s
⊢ NeBot (φ (f ↑x))
case refine'_2
α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
f : α → β
inst✝¹ : LinearOrder β
inst✝ : OrderClosedTopology β
s : Set α
ne_s : Set.Nonempty s
hs : IsCompact s
hf : LowerSemicontinuousOn f s
this✝¹ : Nonempty α
this✝ : Nonempty ↑s
φ : β → Filter α := fun b => 𝓟 (s ∩ f ⁻¹' Iic b)
ℱ : Filter α := ⨅ a, φ (f ↑a)
x : ↑s
this : pure ↑x ≤ φ (f ↑x)
⊢ NeBot (φ (f ↑x))