-- Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity
Goal:
R : Type u_2
inst : CommRing R
m n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] MvPolynomial (Fin m) R
k : ℕ
d : Multiset (Fin m)
S : ConstructibleSetData (MvPolynomial (Fin m) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS : ∀ C ∈ S, ∀ (j : Fin C.n), (C.g j).degrees ≤ d
hf :
  ∀ (i : Fin n),
    degrees.{u_2, 0} (R := R) ((f : (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin m) R) (MvPolynomial.X i)) ≤ d
g : MvPolynomial (Fin m) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R := eval₂Hom f.toRingHom MvPolynomial.X
hg :
  Eq (α := MvPolynomial (Fin n) R →+* MvPolynomial (Fin m) R)
    (g.comp (algebraMap (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))))
    (↑f : MvPolynomial (Fin n) R →+* MvPolynomial (Fin m) R)
σ : MvPolynomial (Fin m) R →+* MvPolynomial (Fin m) (MvPolynomial (Fin n) R) :=
  MvPolynomial.map (algebraMap R (MvPolynomial (Fin n) R))
hσ : g.comp σ = RingHom.id (MvPolynomial (Fin m) R)
hσ' :
  ∀ (x : MvPolynomial (Fin m) R),
    (g : (a : MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) → MvPolynomial (Fin m) R)
        ((σ : (a : MvPolynomial (Fin m) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) x) =
      x
hg' :
  Surjective (α := MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) (β := MvPolynomial (Fin m) R)
    (⇑g : (a : MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) → MvPolynomial (Fin m) R)
S' : ConstructibleSetData (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Finset.image
    (fun (x : BasicConstructibleSetData (MvPolynomial (Fin m) R)) ↦
      match (motive :=
        BasicConstructibleSetData (MvPolynomial (Fin m) R) →
          BasicConstructibleSetData (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)))
        x with
      | { f := fk, n := k, g := gk } =>
        { f := (σ : (a : MvPolynomial (Fin m) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) fk, n := k + n,
          g := fun (s : Fin (k + n)) ↦
            Sum.elim ((⇑σ : (a : MvPolynomial (Fin m) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) ∘ gk)
              (fun (i : Fin n) ↦
                HSub.hSub (α := MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) (β :=
                  MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
                  ((MvPolynomial.C : (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
                      (MvPolynomial.X i) :
                    MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
                  ((σ : (a : MvPolynomial (Fin m) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
                      ((f : (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin m) R) (MvPolynomial.X i)) :
                    MvPolynomial (Fin m) (MvPolynomial (Fin n) R)))
              ((Equiv.symm finSumFinEquiv : (a : Fin (k + n)) → Fin k ⊕ Fin n) s) })
    S
s₀ : Set (MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) :=
  Set.range fun (i : Fin n) ↦
    HSub.hSub (α := MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) (β := MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
      ((MvPolynomial.C : (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
          (MvPolynomial.X i) :
        MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
      ((σ : (a : MvPolynomial (Fin m) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
          ((f : (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin m) R) (MvPolynomial.X i)) :
        MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
hs : zeroLocus s₀ = Set.range (comap g)
hg'' : ∀ (t : Set (PrimeSpectrum (MvPolynomial (Fin m) R))), comap g '' t = comap σ ⁻¹' t ∩ zeroLocus s₀
simp :
  Eq.{u_2 + 1} (α := Set (PrimeSpectrum (MvPolynomial (Fin m) (MvPolynomial (Fin n) R))))
    (⋃ i ∈ S, comap g '' zeroLocus (Set.range i.g) \ comap g '' zeroLocus {i.f})
    (⋃ y ∈ S,
      (zeroLocus
            ((⇑σ : (a : MvPolynomial (Fin m) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) '' Set.range y.g) ∩
          zeroLocus
            (Set.range fun (i : Fin n) ↦
              HSub.hSub (α := MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) (β :=
                MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
                ((MvPolynomial.C : (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
                    (MvPolynomial.X i) :
                  MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
                ((σ : (a : MvPolynomial (Fin m) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
                    ((f : (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin m) R) (MvPolynomial.X i)) :
                  MvPolynomial (Fin m) (MvPolynomial (Fin n) R)))) \
        zeroLocus
          (singleton (α := MvPolynomial (Fin m) (MvPolynomial (Fin n) R))
            ((σ : (a : MvPolynomial (Fin m) R) → MvPolynomial (Fin m) (MvPolynomial (Fin n) R)) y.f :
              MvPolynomial (Fin m) (MvPolynomial (Fin n) R))))
⊢ comap g '' S.toSet = S'.toSet

Tactic:
simp only [S', BasicConstructibleSetData.toSet, ConstructibleSetData.toSet, Set.image_iUnion₂,
      Finset.set_biUnion_finset_image, ← comp_def (g := finSumFinEquiv.symm), Set.range_comp,
      Equiv.range_eq_univ, Set.image_univ, Set.Sum.elim_range,
      Set.image_diff (hf := comap_injective_of_surjective g hg'), zeroLocus_union]

Premises used: Set.range_comp, PrimeSpectrum.BasicConstructibleSetData.toSet, Set.image_univ, Finset.set_biUnion_finset_image, Set.Sum.elim_range, Set.image_diff, PrimeSpectrum.zeroLocus_union, Set.image_iUnion₂, Function.comp_def, finSumFinEquiv, PrimeSpectrum.comap_injective_of_surjective, Equiv.range_eq_univ, Equiv.symm, DFunLike.coe, PrimeSpectrum.ConstructibleSetData.toSet