/-- If `f` is a family of mutually independent random variables (`iIndepFun m f μ`) and `S, T` are
two disjoint finite index sets, then the tuple formed by `f i` for `i ∈ S` is independent of the
tuple `(f i)_i` for `i ∈ T`. -/
theorem iIndepFun.indepFun_finset [IsMarkovKernel κ] (S T : Finset ι) (hST : Disjoint S T)
    (hf_Indep : iIndepFun m f κ μ) (hf_meas : ∀ i, Measurable (f i)) :
    IndepFun (fun a (i : S) => f i a) (fun a (i : T) => f i a) κ μ := by
  -- We introduce π-systems, built from the π-system of boxes which generates `MeasurableSpace.pi`.
  let πSβ := Set.pi (Set.univ : Set S) ''
    Set.pi (Set.univ : Set S) fun i => { s : Set (β i) | MeasurableSet[m i] s }
  let πS := { s : Set Ω | ∃ t ∈ πSβ, (fun a (i : S) => f i a) ⁻¹' t = s }
  have hπS_pi : IsPiSystem πS := by exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _
  have hπS_gen : (MeasurableSpace.pi.comap fun a (i : S) => f i a) = generateFrom πS := by
    rw [generateFrom_pi.symm, comap_generateFrom]
    congr
  let πTβ := Set.pi (Set.univ : Set T) ''
      Set.pi (Set.univ : Set T) fun i => { s : Set (β i) | MeasurableSet[m i] s }
  let πT := { s : Set Ω | ∃ t ∈ πTβ, (fun a (i : T) => f i a) ⁻¹' t = s }
  have hπT_pi : IsPiSystem πT := by exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _
  have hπT_gen : (MeasurableSpace.pi.comap fun a (i : T) => f i a) = generateFrom πT := by
    rw [generateFrom_pi.symm, comap_generateFrom]
    congr
  -- To prove independence, we prove independence of the generating π-systems.
  refine IndepSets.indep (Measurable.comap_le (measurable_pi_iff.mpr fun i => hf_meas i))
    (Measurable.comap_le (measurable_pi_iff.mpr fun i => hf_meas i)) hπS_pi hπT_pi hπS_gen hπT_gen
    ?_
  rintro _ _ ⟨s, ⟨sets_s, hs1, hs2⟩, rfl⟩ ⟨t, ⟨sets_t, ht1, ht2⟩, rfl⟩
  simp only [Set.mem_univ_pi, Set.mem_setOf_eq] at hs1 ht1
  rw [← hs2, ← ht2]
  classical
  let sets_s' : ∀ i : ι, Set (β i) := fun i =>
    dite (i ∈ S) (fun hi => sets_s ⟨i, hi⟩) fun _ => Set.univ
  have h_sets_s'_eq : ∀ {i} (hi : i ∈ S), sets_s' i = sets_s ⟨i, hi⟩ := by
    intro i hi; simp_rw [dif_pos hi]
  have h_sets_s'_univ : ∀ {i} (_hi : i ∈ T), sets_s' i = Set.univ := by
    intro i hi; simp_rw [dif_neg (Finset.disjoint_right.mp hST hi)]
  let sets_t' : ∀ i : ι, Set (β i) := fun i =>
    dite (i ∈ T) (fun hi => sets_t ⟨i, hi⟩) fun _ => Set.univ
  have h_sets_t'_univ : ∀ {i} (_hi : i ∈ S), sets_t' i = Set.univ := by
    intro i hi; simp_rw [dif_neg (Finset.disjoint_left.mp hST hi)]
  have h_meas_s' : ∀ i ∈ S, MeasurableSet (sets_s' i) := by
    intro i hi; rw [h_sets_s'_eq hi]; exact hs1 _
  have h_meas_t' : ∀ i ∈ T, MeasurableSet (sets_t' i) := by
    intro i hi; simp_rw [dif_pos hi]; exact ht1 _
  have h_eq_inter_S : (fun (ω : Ω) (i : ↥S) =>
    f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i := by
    ext1 x
    simp_rw [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
    constructor <;> intro h
    · intro i hi; simp only [h_sets_s'_eq hi, Set.mem_preimage]; exact h ⟨i, hi⟩
    · rintro ⟨i, hi⟩; specialize h i hi; rwa [dif_pos hi] at h
  have h_eq_inter_T : (fun (ω : Ω) (i : ↥T) => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t
    = ⋂ i ∈ T, f i ⁻¹' sets_t' i := by
    ext1 x
    simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
    constructor <;> intro h
    · intro i hi; simp_rw [dif_pos hi]; exact h ⟨i, hi⟩
    · rintro ⟨i, hi⟩; specialize h i hi; simp_rw [dif_pos hi] at h; exact h
  rw [iIndepFun_iff_measure_inter_preimage_eq_mul] at hf_Indep
  have h_Inter_inter :
    ((⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i) =
      ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) := by
    ext1 x
    simp_rw [Set.mem_inter_iff, Set.mem_iInter, Set.mem_preimage, Finset.mem_union]
    constructor <;> intro h
    · intro i hi
      cases' hi with hiS hiT
      · replace h := h.1 i hiS
        simp_rw [dif_pos hiS, dif_neg (Finset.disjoint_left.mp hST hiS)]
        exact ⟨by rwa [dif_pos hiS] at h, Set.mem_univ _⟩
      · replace h := h.2 i hiT
        simp_rw [dif_pos hiT, dif_neg (Finset.disjoint_right.mp hST hiT)]
        exact ⟨Set.mem_univ _, by rwa [dif_pos hiT] at h⟩
    · exact ⟨fun i hi => (h i (Or.inl hi)).1, fun i hi => (h i (Or.inr hi)).2⟩
  have h_meas_inter : ∀ i ∈ S ∪ T, MeasurableSet (sets_s' i ∩ sets_t' i) := by
    intros i hi_mem
    rw [Finset.mem_union] at hi_mem
    cases' hi_mem with hi_mem hi_mem
    · rw [h_sets_t'_univ hi_mem, Set.inter_univ]
      exact h_meas_s' i hi_mem
    · rw [h_sets_s'_univ hi_mem, Set.univ_inter]
      exact h_meas_t' i hi_mem
  filter_upwards [hf_Indep S h_meas_s', hf_Indep T h_meas_t', hf_Indep (S ∪ T) h_meas_inter]
    with a h_indepS h_indepT h_indepST -- todo: this unfolded sets_s', sets_t'?
  

Proof state:
case h
α : Type u_1
Ω : Type u_2
ι : Type u_3
β✝ : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : ↥(kernel α Ω)
μ : Measure α
f✝ : Ω → β✝
g : Ω → β'
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
inst✝ : IsMarkovKernel κ
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) →
      ∀ᵐ (a : α) ∂μ, ↑↑(κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i in S, ↑↑(κ a) (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t ∈ πSβ, (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t ∈ πTβ, (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ i ∈ S, MeasurableSet (sets_s' i)
h_meas_t' : ∀ i ∈ T, MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ i ∈ S, f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ i ∈ T, f i ⁻¹' sets_t' i
h_Inter_inter : (⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i = ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)
h_meas_inter : ∀ i ∈ S ∪ T, MeasurableSet (sets_s' i ∩ sets_t' i)
a : α
h_indepS : ↑↑(κ a) (⋂ i ∈ S, f i ⁻¹' sets_s' i) = ∏ i in S, ↑↑(κ a) (f i ⁻¹' sets_s' i)
h_indepT : ↑↑(κ a) (⋂ i ∈ T, f i ⁻¹' sets_t' i) = ∏ i in T, ↑↑(κ a) (f i ⁻¹' sets_t' i)
h_indepST :
  ↑↑(κ a) (⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ∏ i in S ∪ T, ↑↑(κ a) (f i ⁻¹' (sets_s' i ∩ sets_t' i))
⊢ ↑↑(κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) =
    ↑↑(κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) *
      ↑↑(κ a) ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t)

Next tactic:
rw [h_eq_inter_S, h_eq_inter_T, h_indepS, h_indepT, h_Inter_inter, h_indepST,
    Finset.prod_union hST]