-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
theorem measurablySeparable_range_of_disjoint [T2Space α] [MeasurableSpace α]
    [OpensMeasurableSpace α] {f g : (ℕ → ℕ) → α} (hf : Continuous f) (hg : Continuous g)
    (h : Disjoint (range f) (range g)) : MeasurablySeparable (range f) (range g)

Proof state:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous f
hg : Continuous g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
n : ℕ
x y : ℕ → ℕ
H :
  ∀ (x' y' : ℕ → ℕ),
    x' ∈ cylinder x n → y' ∈ cylinder y n → MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
⊢ MeasurablySeparable (⋃ i, f '' cylinder (update x n i) (n + 1)) (⋃ i, g '' cylinder (update y n i) (n + 1))

Tactic:
refine MeasurablySeparable.iUnion fun i j => ?_

Result:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous f
hg : Continuous g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
n : ℕ
x y : ℕ → ℕ
H :
  ∀ (x' y' : ℕ → ℕ),
    x' ∈ cylinder x n → y' ∈ cylinder y n → MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
i j : ℕ
⊢ MeasurablySeparable (f '' cylinder (update x n i) (n + 1)) (g '' cylinder (update y n j) (n + 1))

Proof state:
case succ
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous f
hg : Continuous g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x => 0, fun x => 0), ⋯⟩
p : ℕ → A := fun n => F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
m n : ℕ
hmn : m + 1 ≤ n
IH : (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
⊢ (↑(p (n + 1))).2.2 m = (↑(p (m + 1))).2.2 m

Tactic:
have I : (F (p n)).val.snd.snd m = (p n).val.snd.snd m := by
  apply hFy (p n) m
  rw [pn_fst]
  exact hmn

Result:
case succ
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous f
hg : Continuous g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I✝ :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x => 0, fun x => 0), ⋯⟩
p : ℕ → A := fun n => F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
m n : ℕ
hmn : m + 1 ≤ n
IH : (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
I : (↑(F (p n))).2.2 m = (↑(p n)).2.2 m
⊢ (↑(p (n + 1))).2.2 m = (↑(p (m + 1))).2.2 m

Proof state:
case h.e'_1.h.e'_3.h.e'_4
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous f
hg : Continuous g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x => 0, fun x => 0), ⋯⟩
p : ℕ → A := fun n => F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
Iy : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
x : ℕ → ℕ := fun n => (↑(p (n + 1))).2.1 n
hx : x = fun n => (↑(p (n + 1))).2.1 n
y : ℕ → ℕ := fun n => (↑(p (n + 1))).2.2 n
hy : y = fun n => (↑(p (n + 1))).2.2 n
n : ℕ
⊢ ∀ i < n, x i = (↑(p n)).2.1 i

Tactic:
intro i hi

Result:
case h.e'_1.h.e'_3.h.e'_4
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : T2Space α
inst✝¹ : MeasurableSpace α
inst✝ : OpensMeasurableSpace α
f g : (ℕ → ℕ) → α
hf : Continuous f
hg : Continuous g
h : Disjoint (range f) (range g)
hfg : ¬MeasurablySeparable (range f) (range g)
I :
  ∀ (n : ℕ) (x y : ℕ → ℕ),
    ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →
      ∃ x' y',
        x' ∈ cylinder x n ∧
          y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))
A : Type := { p // ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }
F : A → A
hFn : ∀ (p : A), (↑(F p)).1 = (↑p).1 + 1
hFx : ∀ (p : A), (↑(F p)).2.1 ∈ cylinder (↑p).2.1 (↑p).1
hFy : ∀ (p : A), (↑(F p)).2.2 ∈ cylinder (↑p).2.2 (↑p).1
p0 : A := ⟨(0, fun x => 0, fun x => 0), ⋯⟩
p : ℕ → A := fun n => F^[n] p0
prec : ∀ (n : ℕ), p (n + 1) = F (p n)
pn_fst : ∀ (n : ℕ), (↑(p n)).1 = n
Ix : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.1 m = (↑(p (m + 1))).2.1 m
Iy : ∀ (m n : ℕ), m + 1 ≤ n → (↑(p n)).2.2 m = (↑(p (m + 1))).2.2 m
x : ℕ → ℕ := fun n => (↑(p (n + 1))).2.1 n
hx : x = fun n => (↑(p (n + 1))).2.1 n
y : ℕ → ℕ := fun n => (↑(p (n + 1))).2.2 n
hy : y = fun n => (↑(p (n + 1))).2.2 n
n i : ℕ
hi : i < n
⊢ x i = (↑(p n)).2.1 i