simp [A, Set.ext_iff] at heq
x : ↑N0
A : Set ↑N0 := {x | mul (S z, x) = x}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq : Subtype.val '' A = N0
⊢ mul (S z, x) = x
x : ↑N0
A : Set ↑N0 := {x | mul (S z, x) = x}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq :
  ∀ (x : α),
    (∃ (x_1 : x ∈ N0), mul (S z, { val := x, property := (_ : x ∈ N0) }) = { val := x, property := (_ : x ∈ N0) }) ↔
      x ∈ N0
⊢ mul (S z, x) = x