Theorem: MeasurableSet.analyticSet_image
File: Mathlib/MeasureTheory/Constructions/Polish/Basic.lean

Proof steps:
state_before: α : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α
X : Type u_3
Y : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : StandardBorelSpace X
inst✝³ : TopologicalSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : OpensMeasurableSpace Y
f : X → Y
inst✝ : SecondCountableTopology ↑(range f)
s : Set X
hs : MeasurableSet s
hf : Measurable f
⊢ AnalyticSet (f '' s)
state_after: α : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α
X : Type u_3
Y : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : StandardBorelSpace X
inst✝³ : TopologicalSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : OpensMeasurableSpace Y
f : X → Y
inst✝ : SecondCountableTopology ↑(range f)
s : Set X
hs : MeasurableSet s
hf : Measurable f
this : UpgradedStandardBorel X := upgradeStandardBorel X
⊢ AnalyticSet (f '' s)
tactic: letI := upgradeStandardBorel X
state_before: α : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α
X : Type u_3
Y : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : StandardBorelSpace X
inst✝³ : TopologicalSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : OpensMeasurableSpace Y
f : X → Y
inst✝ : SecondCountableTopology ↑(range f)
s : Set X
hs : MeasurableSet s
hf : Measurable f
this : UpgradedStandardBorel X := upgradeStandardBorel X
⊢ AnalyticSet (f '' s)
state_after: case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α
X : Type u_3
Y : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : StandardBorelSpace X
inst✝³ : TopologicalSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : OpensMeasurableSpace Y
f : X → Y
inst✝ : SecondCountableTopology ↑(range f)
s : Set X
hs : MeasurableSet s
hf : Measurable f
this : UpgradedStandardBorel X := upgradeStandardBorel X
τ' : TopologicalSpace X
hle : τ' ≤ UpgradedStandardBorel.toTopologicalSpace
hfc : Continuous f
hτ' : PolishSpace X
⊢ AnalyticSet (f '' s)
tactic: rcases hf.exists_continuous with ⟨τ', hle, hfc, hτ'⟩
state_before: case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α
X : Type u_3
Y : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : StandardBorelSpace X
inst✝³ : TopologicalSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : OpensMeasurableSpace Y
f : X → Y
inst✝ : SecondCountableTopology ↑(range f)
s : Set X
hs : MeasurableSet s
hf : Measurable f
this : UpgradedStandardBorel X := upgradeStandardBorel X
τ' : TopologicalSpace X
hle : τ' ≤ UpgradedStandardBorel.toTopologicalSpace
hfc : Continuous f
hτ' : PolishSpace X
⊢ AnalyticSet (f '' s)
state_after: case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α
X : Type u_3
Y : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : StandardBorelSpace X
inst✝³ : TopologicalSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : OpensMeasurableSpace Y
f : X → Y
inst✝ : SecondCountableTopology ↑(range f)
s : Set X
hs : MeasurableSet s
hf : Measurable f
this : UpgradedStandardBorel X := upgradeStandardBorel X
τ' : TopologicalSpace X
hle : τ' ≤ UpgradedStandardBorel.toTopologicalSpace
hfc : Continuous f
hτ' : PolishSpace X
m' : MeasurableSpace X := borel X
⊢ AnalyticSet (f '' s)
tactic: letI m' : MeasurableSpace X := @borel _ τ'
state_before: case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α
X : Type u_3
Y : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : StandardBorelSpace X
inst✝³ : TopologicalSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : OpensMeasurableSpace Y
f : X → Y
inst✝ : SecondCountableTopology ↑(range f)
s : Set X
hs : MeasurableSet s
hf : Measurable f
this : UpgradedStandardBorel X := upgradeStandardBorel X
τ' : TopologicalSpace X
hle : τ' ≤ UpgradedStandardBorel.toTopologicalSpace
hfc : Continuous f
hτ' : PolishSpace X
m' : MeasurableSpace X := borel X
⊢ AnalyticSet (f '' s)
state_after: case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α
X : Type u_3
Y : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : StandardBorelSpace X
inst✝³ : TopologicalSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : OpensMeasurableSpace Y
f : X → Y
inst✝ : SecondCountableTopology ↑(range f)
s : Set X
hs : MeasurableSet s
hf : Measurable f
this : UpgradedStandardBorel X := upgradeStandardBorel X
τ' : TopologicalSpace X
hle : τ' ≤ UpgradedStandardBorel.toTopologicalSpace
hfc : Continuous f
hτ' : PolishSpace X
m' : MeasurableSpace X := borel X
b' : BorelSpace X
⊢ AnalyticSet (f '' s)
tactic: haveI b' : BorelSpace X := ⟨rfl⟩
state_before: case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α
X : Type u_3
Y : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : StandardBorelSpace X
inst✝³ : TopologicalSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : OpensMeasurableSpace Y
f : X → Y
inst✝ : SecondCountableTopology ↑(range f)
s : Set X
hs : MeasurableSet s
hf : Measurable f
this : UpgradedStandardBorel X := upgradeStandardBorel X
τ' : TopologicalSpace X
hle : τ' ≤ UpgradedStandardBorel.toTopologicalSpace
hfc : Continuous f
hτ' : PolishSpace X
m' : MeasurableSpace X := borel X
b' : BorelSpace X
⊢ AnalyticSet (f '' s)
state_after: case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α
X : Type u_3
Y : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : StandardBorelSpace X
inst✝³ : TopologicalSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : OpensMeasurableSpace Y
f : X → Y
inst✝ : SecondCountableTopology ↑(range f)
s : Set X
hs : MeasurableSet s
hf : Measurable f
this : UpgradedStandardBorel X := upgradeStandardBorel X
τ' : TopologicalSpace X
hle✝ : τ' ≤ UpgradedStandardBorel.toTopologicalSpace
hfc : Continuous f
hτ' : PolishSpace X
m' : MeasurableSpace X := borel X
b' : BorelSpace X
hle : borel X ≤ borel X
⊢ AnalyticSet (f '' s)
tactic: have hle := borel_anti hle
state_before: case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁶ : TopologicalSpace α
X : Type u_3
Y : Type u_4
inst✝⁵ : MeasurableSpace X
inst✝⁴ : StandardBorelSpace X
inst✝³ : TopologicalSpace Y
inst✝² : MeasurableSpace Y
inst✝¹ : OpensMeasurableSpace Y
f : X → Y
inst✝ : SecondCountableTopology ↑(range f)
s : Set X
hs : MeasurableSet s
hf : Measurable f
this : UpgradedStandardBorel X := upgradeStandardBorel X
τ' : TopologicalSpace X
hle✝ : τ' ≤ UpgradedStandardBorel.toTopologicalSpace
hfc : Continuous f
hτ' : PolishSpace X
m' : MeasurableSpace X := borel X
b' : BorelSpace X
hle : borel X ≤ borel X
⊢ AnalyticSet (f '' s)
state_after: no goals
tactic: exact (hle _ hs).analyticSet.image_of_continuous hfc