apply Iff.intro
α : Type u_1
β : Type u_2
f : α → β
b : β
x : α
xs : List α
ih : b ∈ map f xs ↔ ∃ a, a ∈ xs ∧ f a = b
⊢ b ∈ map f (x :: xs) ↔ ∃ a, a ∈ x :: xs ∧ f a = b
case mp
α : Type u_1
β : Type u_2
f : α → β
b : β
x : α
xs : List α
ih : b ∈ map f xs ↔ ∃ a, a ∈ xs ∧ f a = b
⊢ b ∈ map f (x :: xs) → ∃ a, a ∈ x :: xs ∧ f a = b

case mpr
α : Type u_1
β : Type u_2
f : α → β
b : β
x : α
xs : List α
ih : b ∈ map f xs ↔ ∃ a, a ∈ xs ∧ f a = b
⊢ (∃ a, a ∈ x :: xs ∧ f a = b) → b ∈ map f (x :: xs)