. exact fun a => ⟨g a, hfg a⟩
case refine_1
α : Type u
β : Type v
inst✝³ : BishopFin α
inst✝² : BishopFin β
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
f : β → α
g : α → β
hgf : ∀ (b : β), g (f b) = b
hfg : ∀ (a : α), f (g a) = a
⊢ ∀ (a : α), ∃ b, f b = a

case refine_2
α : Type u
β : Type v
inst✝³ : BishopFin α
inst✝² : BishopFin β
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
f : β → α
g : α → β
hgf : ∀ (b : β), g (f b) = b
hfg : ∀ (a : α), f (g a) = a
⊢ ∀ (b₁ b₂ : β), f b₁ = f b₂ → b₁ = b₂
case refine_2
α : Type u
β : Type v
inst✝³ : BishopFin α
inst✝² : BishopFin β
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
f : β → α
g : α → β
hgf : ∀ (b : β), g (f b) = b
hfg : ∀ (a : α), f (g a) = a
⊢ ∀ (b₁ b₂ : β), f b₁ = f b₂ → b₁ = b₂