English
Membership in a restricted part equals conjunction of the condition and membership in the ambient part.
Русский
Членство в ограниченной части равно конъюнкции условия и членство в внешнем множестве.
LaTeX
$$$a \\in \\restrict p\\ o\\ h \\leftrightarrow p \\land a \\in o$.$$
Lean4
@[simp]
theorem mem_restrict (p : Prop) (o : Part α) (h : p → o.Dom) (a : α) : a ∈ restrict p o h ↔ p ∧ a ∈ o :=
by
dsimp [restrict, mem_eq]; constructor
· rintro ⟨h₀, h₁⟩
exact ⟨h₀, ⟨_, h₁⟩⟩
rintro ⟨h₀, _, h₂⟩; exact ⟨h₀, h₂⟩