-- Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
theorem measurableSet_range_of_continuous_injective {β : Type*} [TopologicalSpace γ]
    [PolishSpace γ] [TopologicalSpace β] [T2Space β] [MeasurableSpace β] [OpensMeasurableSpace β]
    {f : γ → β} (f_cont : Continuous f) (f_inj : Injective f) :
    MeasurableSet (range f)

Proof state:
case intro.intro.intro.intro.intro.intro.h₂
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous f
f_inj : Injective f
this✝ : UpgradedPolishSpace γ := upgradePolishSpace γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s => closure (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n => ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
x : β
hx : x ∈ ⋂ n, F n
s : ℕ → ↑b
hs : ∀ (n : ℕ), Bornology.IsBounded ↑(s n) ∧ diam ↑(s n) ≤ u n
hxs : ∀ (n : ℕ), x ∈ E (s n)
C2 : ∀ (n : ℕ), (↑(s n)).Nonempty
⊢ x ∈ range f

Tactic:
choose y hy using C2

Result:
case intro.intro.intro.intro.intro.intro.h₂
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous f
f_inj : Injective f
this✝ : UpgradedPolishSpace γ := upgradePolishSpace γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s => closure (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n => ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
x : β
hx : x ∈ ⋂ n, F n
s : ℕ → ↑b
hs : ∀ (n : ℕ), Bornology.IsBounded ↑(s n) ∧ diam ↑(s n) ≤ u n
hxs : ∀ (n : ℕ), x ∈ E (s n)
y : ℕ → γ
hy : ∀ (n : ℕ), y n ∈ ↑(s n)
⊢ x ∈ range f

Proof state:
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous f
f_inj : Injective f
this✝ : UpgradedPolishSpace γ := upgradePolishSpace γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s => closure (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n => ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
x : β
hx : x ∈ ⋂ n, F n
s : ℕ → ↑b
hs : ∀ (n : ℕ), Bornology.IsBounded ↑(s n) ∧ diam ↑(s n) ≤ u n
hxs : ∀ (n : ℕ), x ∈ E (s n)
y : ℕ → γ
hy : ∀ (n : ℕ), y n ∈ ↑(s n)
m n : ℕ
⊢ (↑(s m) ∩ ↑(s n)).Nonempty

Tactic:
rw [← not_disjoint_iff_nonempty_inter]

Result:
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous f
f_inj : Injective f
this✝ : UpgradedPolishSpace γ := upgradePolishSpace γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s => closure (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n => ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
x : β
hx : x ∈ ⋂ n, F n
s : ℕ → ↑b
hs : ∀ (n : ℕ), Bornology.IsBounded ↑(s n) ∧ diam ↑(s n) ≤ u n
hxs : ∀ (n : ℕ), x ∈ E (s n)
y : ℕ → γ
hy : ∀ (n : ℕ), y n ∈ ↑(s n)
m n : ℕ
⊢ ¬Disjoint ↑(s m) ↑(s n)

Proof state:
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous f
f_inj : Injective f
this✝ : UpgradedPolishSpace γ := upgradePolishSpace γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
⊢ ∀ (p : A), ∃ q, f '' ↑(↑p).1 ⊆ q ∧ Disjoint (f '' ↑(↑p).2) q ∧ MeasurableSet q

Tactic:
intro p

Result:
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous f
f_inj : Injective f
this✝ : UpgradedPolishSpace γ := upgradePolishSpace γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
p : A
⊢ ∃ q, f '' ↑(↑p).1 ⊆ q ∧ Disjoint (f '' ↑(↑p).2) q ∧ MeasurableSet q

Proof state:
case intro.intro.intro.intro.intro.intro.h₁.intro.intro.intro.intro.refine_1.a
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous f
f_inj : Injective f
this✝ : UpgradedPolishSpace γ := upgradePolishSpace γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s => closure (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n => ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
y : γ
n : ℕ
s : Set γ
sb : s ∈ b
ys : y ∈ s
hs : s ⊆ ball y (u n / 2)
diam_s : diam s ≤ u n
t : ↑b
ht : Disjoint ↑⟨s, sb⟩ ↑t
⊢ f y ∈ f '' ↑(↑⟨(⟨s, sb⟩, t), ht⟩).1

Tactic:
exact mem_image_of_mem _ ys

Result:
no goals