-- Module: Mathlib.AlgebraicGeometry.Morphisms.Basic
Goal:
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst : HasAffineProperty P Q
X_1 Y_1 : Scheme
f_1 : X_1 ⟶ Y_1
this : Q.IsLocal := ⋯
X Y : Scheme
f : X ⟶ Y
ι : Type u_1
U : ι → Y.Opens
hU : iSup U = ⊤
H : ∀ (i : ι), P (X := (↑(f ⁻¹ᵁ U i) : Scheme)) (f ∣_ U i)
𝒰 : Y.OpenCover := Y.openCoverOfIsOpenCover U hU
i : 𝒰.I₀
j :
  PreZeroHypercover.I₀ (C := Scheme) (S := 𝒰.X i)
    ((fun (i : 𝒰.I₀) ↦
        (Precoverage.ZeroHypercover.toPreZeroHypercover (J := Scheme.precoverage IsOpenImmersion)
            ((fun (j : 𝒰.I₀) ↦ ((𝒰.X j).affineCover : (𝒰.X j).OpenCover)) i) :
          PreZeroHypercover (𝒰.X i)))
      i)
of_sSup_eq_top :
  ∀
    (this :
      P (X := (Precoverage.ZeroHypercover.pullback₁ f 𝒰).X i)
        (Scheme.Cover.pullbackHom (P := IsOpenImmersion) 𝒰 f i :
          (Precoverage.ZeroHypercover.pullback₁ f 𝒰).X i ⟶ 𝒰.X i)),
    Q (X :=
      (Precoverage.ZeroHypercover.pullback₁ f 𝒰.affineRefinement.openCover).X
        (Sigma.mk (β := fun (i : 𝒰.I₀) ↦
          PreZeroHypercover.I₀ (C := Scheme) (S := 𝒰.X i)
            ((fun (i : 𝒰.I₀) ↦
                (Precoverage.ZeroHypercover.toPreZeroHypercover (J := Scheme.precoverage IsOpenImmersion)
                    ((fun (j : 𝒰.I₀) ↦ ((𝒰.X j).affineCover : (𝒰.X j).OpenCover)) i) :
                  PreZeroHypercover (𝒰.X i)))
              i))
          i j))
      (Scheme.Cover.pullbackHom (P := IsOpenImmersion) 𝒰.affineRefinement.openCover f
          (Sigma.mk (β := fun (i : 𝒰.I₀) ↦
            PreZeroHypercover.I₀ (C := Scheme) (S := 𝒰.X i)
              ((fun (i : 𝒰.I₀) ↦
                  (Precoverage.ZeroHypercover.toPreZeroHypercover (J := Scheme.precoverage IsOpenImmersion)
                      ((fun (j : 𝒰.I₀) ↦ ((𝒰.X j).affineCover : (𝒰.X j).OpenCover)) i) :
                    PreZeroHypercover (𝒰.X i)))
                i))
            i j) :
        (Precoverage.ZeroHypercover.pullback₁ f 𝒰.affineRefinement.openCover).X
            (Sigma.mk (β := fun (i : 𝒰.I₀) ↦
              PreZeroHypercover.I₀ (C := Scheme) (S := 𝒰.X i)
                ((fun (i : 𝒰.I₀) ↦
                    (Precoverage.ZeroHypercover.toPreZeroHypercover (J := Scheme.precoverage IsOpenImmersion)
                        ((fun (j : 𝒰.I₀) ↦ ((𝒰.X j).affineCover : (𝒰.X j).OpenCover)) i) :
                      PreZeroHypercover (𝒰.X i)))
                  i))
              i j) ⟶
          𝒰.affineRefinement.openCover.X
            (Sigma.mk (β := fun (i : 𝒰.I₀) ↦
              PreZeroHypercover.I₀ (C := Scheme) (S := 𝒰.X i)
                ((fun (i : 𝒰.I₀) ↦
                    (Precoverage.ZeroHypercover.toPreZeroHypercover (J := Scheme.precoverage IsOpenImmersion)
                        ((fun (j : 𝒰.I₀) ↦ ((𝒰.X j).affineCover : (𝒰.X j).OpenCover)) i) :
                      PreZeroHypercover (𝒰.X i)))
                  i))
              i j))
⊢ Q (X :=
    (Precoverage.ZeroHypercover.pullback₁ f 𝒰.affineRefinement.openCover).X
      (Sigma.mk (β := fun (i : 𝒰.I₀) ↦
        PreZeroHypercover.I₀ (C := Scheme) (S := 𝒰.X i)
          ((fun (i : 𝒰.I₀) ↦
              (Precoverage.ZeroHypercover.toPreZeroHypercover (J := Scheme.precoverage IsOpenImmersion)
                  ((fun (j : 𝒰.I₀) ↦ ((𝒰.X j).affineCover : (𝒰.X j).OpenCover)) i) :
                PreZeroHypercover (𝒰.X i)))
            i))
        i j))
    (Scheme.Cover.pullbackHom (P := IsOpenImmersion) 𝒰.affineRefinement.openCover f
        (Sigma.mk (β := fun (i : 𝒰.I₀) ↦
          PreZeroHypercover.I₀ (C := Scheme) (S := 𝒰.X i)
            ((fun (i : 𝒰.I₀) ↦
                (Precoverage.ZeroHypercover.toPreZeroHypercover (J := Scheme.precoverage IsOpenImmersion)
                    ((fun (j : 𝒰.I₀) ↦ ((𝒰.X j).affineCover : (𝒰.X j).OpenCover)) i) :
                  PreZeroHypercover (𝒰.X i)))
              i))
          i j) :
      (Precoverage.ZeroHypercover.pullback₁ f 𝒰.affineRefinement.openCover).X
          (Sigma.mk (β := fun (i : 𝒰.I₀) ↦
            PreZeroHypercover.I₀ (C := Scheme) (S := 𝒰.X i)
              ((fun (i : 𝒰.I₀) ↦
                  (Precoverage.ZeroHypercover.toPreZeroHypercover (J := Scheme.precoverage IsOpenImmersion)
                      ((fun (j : 𝒰.I₀) ↦ ((𝒰.X j).affineCover : (𝒰.X j).OpenCover)) i) :
                    PreZeroHypercover (𝒰.X i)))
                i))
            i j) ⟶
        𝒰.affineRefinement.openCover.X
          (Sigma.mk (β := fun (i : 𝒰.I₀) ↦
            PreZeroHypercover.I₀ (C := Scheme) (S := 𝒰.X i)
              ((fun (i : 𝒰.I₀) ↦
                  (Precoverage.ZeroHypercover.toPreZeroHypercover (J := Scheme.precoverage IsOpenImmersion)
                      ((fun (j : 𝒰.I₀) ↦ ((𝒰.X j).affineCover : (𝒰.X j).OpenCover)) i) :
                    PreZeroHypercover (𝒰.X i)))
                i))
            i j))

Tactic:
have : P (𝒰.pullbackHom f i) := by
      refine (P.arrow_mk_iso_iff
        (morphismRestrictEq _ ?_ ≪≫ morphismRestrictOpensRange f (𝒰.f i))).mp (H i)
      exact (Scheme.Opens.opensRange_ι _).symm