Theorem: MeasurableEmbedding.borelSpace
File: Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean

Proof steps:
state_before: α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹³ : TopologicalSpace α✝
inst✝¹² : MeasurableSpace α✝
inst✝¹¹ : BorelSpace α✝
inst✝¹⁰ : TopologicalSpace β✝
inst✝⁹ : MeasurableSpace β✝
inst✝⁸ : BorelSpace β✝
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
α : Type u_6
β : Type u_7
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace β
inst✝ : TopologicalSpace β
hβ : BorelSpace β
e : α → β
h'e : MeasurableEmbedding e
h''e : Inducing e
⊢ BorelSpace α
state_after: case measurable_eq
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹³ : TopologicalSpace α✝
inst✝¹² : MeasurableSpace α✝
inst✝¹¹ : BorelSpace α✝
inst✝¹⁰ : TopologicalSpace β✝
inst✝⁹ : MeasurableSpace β✝
inst✝⁸ : BorelSpace β✝
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
α : Type u_6
β : Type u_7
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace β
inst✝ : TopologicalSpace β
hβ : BorelSpace β
e : α → β
h'e : MeasurableEmbedding e
h''e : Inducing e
⊢ inst✝³ = borel α
tactic: constructor
state_before: case measurable_eq
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹³ : TopologicalSpace α✝
inst✝¹² : MeasurableSpace α✝
inst✝¹¹ : BorelSpace α✝
inst✝¹⁰ : TopologicalSpace β✝
inst✝⁹ : MeasurableSpace β✝
inst✝⁸ : BorelSpace β✝
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
α : Type u_6
β : Type u_7
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace β
inst✝ : TopologicalSpace β
hβ : BorelSpace β
e : α → β
h'e : MeasurableEmbedding e
h''e : Inducing e
⊢ inst✝³ = borel α
state_after: case measurable_eq
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹³ : TopologicalSpace α✝
inst✝¹² : MeasurableSpace α✝
inst✝¹¹ : BorelSpace α✝
inst✝¹⁰ : TopologicalSpace β✝
inst✝⁹ : MeasurableSpace β✝
inst✝⁸ : BorelSpace β✝
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
α : Type u_6
β : Type u_7
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace β
inst✝ : TopologicalSpace β
hβ : BorelSpace β
e : α → β
h'e : MeasurableEmbedding e
h''e : Inducing e
this : MeasurableSpace.comap e (borel β) = inst✝³
⊢ inst✝³ = borel α
tactic: have : MeasurableSpace.comap e (borel β) = ‹_› := by simpa [hβ.measurable_eq] using h'e.comap_eq
state_before: case measurable_eq
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α✝
inst✝¹³ : TopologicalSpace α✝
inst✝¹² : MeasurableSpace α✝
inst✝¹¹ : BorelSpace α✝
inst✝¹⁰ : TopologicalSpace β✝
inst✝⁹ : MeasurableSpace β✝
inst✝⁸ : BorelSpace β✝
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace γ
inst✝⁵ : BorelSpace γ
inst✝⁴ : MeasurableSpace δ
α : Type u_6
β : Type u_7
inst✝³ : MeasurableSpace α
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace β
inst✝ : TopologicalSpace β
hβ : BorelSpace β
e : α → β
h'e : MeasurableEmbedding e
h''e : Inducing e
this : MeasurableSpace.comap e (borel β) = inst✝³
⊢ inst✝³ = borel α
state_after: no goals
tactic: rw [← this, ← borel_comap, h''e.induced]