-- Mathlib/Probability/ConditionalProbability.lean
lemma comap_cond {i : Ω' → Ω} (hi : MeasurableEmbedding i) (hi' : ∀ᵐ ω ∂μ, ω ∈ range i)
    (hs : MeasurableSet s) : comap i μ[|s] = (comap i μ)[|i in s]

Proof state:
case h.hms
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ MeasurableSet (i ⁻¹' s)

Tactic:
exact hi.measurable hs

Result:
no goals

Proof state:
case h
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hi' : ∀ᵐ (ω : Ω) ∂μ, ω ∈ range i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
⊢ (comap i μ[|s]) t = (comap i μ)[t | i ⁻¹' s]

Tactic:
change μ (range i)ᶜ = 0 at hi'

Result:
case h
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ (comap i μ[|s]) t = (comap i μ)[t | i ⁻¹' s]

Proof state:
case h.hs
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ MeasurableSet t

Tactic:
exact ht

Result:
no goals

Proof state:
case h.hfi
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ Function.Injective i

Tactic:
exact hi.injective

Result:
no goals

Proof state:
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hi' : ∀ᵐ (ω : Ω) ∂μ, ω ∈ range i
hs : MeasurableSet s
⊢ comap i μ[|s] = (comap i μ)[|i ⁻¹' s]

Tactic:
ext t ht

Result:
case h
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hi' : ∀ᵐ (ω : Ω) ∂μ, ω ∈ range i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
⊢ (comap i μ[|s]) t = (comap i μ)[t | i ⁻¹' s]

Proof state:
case h
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ Function.Injective i

case h.hfi
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ Function.Injective i

case h.hf
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ ∀ (s : Set Ω'), MeasurableSet s → MeasurableSet (i '' s)

case h.hs
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ MeasurableSet (i ⁻¹' s ∩ t)

case h.hfi
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ Function.Injective i

case h.hf
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ ∀ (s : Set Ω'), MeasurableSet s → MeasurableSet (i '' s)

case h.hs
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ MeasurableSet (i ⁻¹' s)

case h.hms
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ MeasurableSet s

case h.hfi
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ Function.Injective i

case h.hf
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ ∀ (s : Set Ω'), MeasurableSet s → MeasurableSet (i '' s)

case h.hs
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ MeasurableSet t

case h.hms
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ MeasurableSet (i ⁻¹' s)

Tactic:
all_goals first
| exact hi.injective
| exact hi.measurableSet_image'
| exact hs
| exact ht
| exact hi.measurable hs
| exact (hi.measurable hs).inter ht

Result:
no goals

Proof state:
case h.hs
Ω : Type u_1
Ω' : Type u_2
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s : Set Ω
i : Ω' → Ω
hi : MeasurableEmbedding i
hs : MeasurableSet s
t : Set Ω'
ht : MeasurableSet t
hi' : μ (range i)ᶜ = 0
⊢ MeasurableSet (i ⁻¹' s ∩ t)

Tactic:
exact (hi.measurable hs).inter ht

Result:
no goals