∀ {ι : Type u_1} {α : Type u_2} {s : ι → Set α} (eq_of_mem : ∀ {x : α} {i j : ι}, x ∈ s i → x ∈ s j → i = j)
  (some : ι → α) (some_mem : ∀ (i : ι), some i ∈ s i) (index : α → ι) (mem_index : ∀ (x : α), x ∈ s (index x))
  (eq_of_mem_1 : ∀ {x : α} {i j : ι}, x ∈ s i → x ∈ s j → i = j) (some_1 : ι → α)
  (some_mem_1 : ∀ (i : ι), some_1 i ∈ s i) (index_1 : α → ι) (mem_index_1 : ∀ (x : α), x ∈ s (index_1 x)),
  ({ eq_of_mem := eq_of_mem, some := some, some_mem := some_mem, index := index, mem_index := mem_index } =
      { eq_of_mem := eq_of_mem_1, some := some_1, some_mem := some_mem_1, index := index_1,
        mem_index := mem_index_1 }) =
    (some = some_1 ∧ index = index_1)