apply Iff.intro
α : Type u_1
f : Filter α
s t : Set α
⊢ s ∈ f ⊓ 𝓟 t ↔ {x | x ∈ t → x ∈ s} ∈ f
case mp
α : Type u_1
f : Filter α
s t : Set α
⊢ s ∈ f ⊓ 𝓟 t → {x | x ∈ t → x ∈ s} ∈ f

case mpr
α : Type u_1
f : Filter α
s t : Set α
⊢ {x | x ∈ t → x ∈ s} ∈ f → s ∈ f ⊓ 𝓟 t