-- Lean 4 proof step from Mathlib.Order.Category.PartOrdEmb (goal)
Goal:
J : Type u
inst : SmallCategory J
inst_1 : IsFiltered J
F : J ⥤ PartOrdEmb
c :
  Cocone (F ⋙ forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier))
hc : IsColimit c
s : Cocone F
j : J
x' y' :
  (F ⋙ forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier)).obj j
hx :
  Eq (α := ToType (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier) s.pt)
    (((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
              (hc.desc
                (Functor.mapCocone
                  (forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC :=
                    carrier))
                  s)) :
            TypeCat.Fun c.pt
              (ToType (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier) s.pt)) :
          (a : c.pt) → ToType s.pt)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) : (a : ToType (F.obj j)) → c.pt) x' : c.pt) :
      ToType (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier) s.pt)
    ((ConcreteCategory.hom (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier)
            (s.ι.app j) :
          (a : (↑(F.obj j) : Type u)) → (↑s.pt : Type u))
        x' :
      (↑s.pt : Type u))
hy :
  Eq (α := ToType (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier) s.pt)
    (((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
              (hc.desc
                (Functor.mapCocone
                  (forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC :=
                    carrier))
                  s)) :
            TypeCat.Fun c.pt
              (ToType (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier) s.pt)) :
          (a : c.pt) → ToType s.pt)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) : (a : ToType (F.obj j)) → c.pt) y' : c.pt) :
      ToType (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier) s.pt)
    ((ConcreteCategory.hom (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier)
            (s.ι.app j) :
          (a : (↑(F.obj j) : Type u)) → (↑s.pt : Type u))
        y' :
      (↑s.pt : Type u))
⊢ LE.le (α := (↑s.pt : Type u))
      (((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
                (hc.desc
                  (Functor.mapCocone
                    (forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC :=
                      carrier))
                    s)) :
              TypeCat.Fun c.pt
                (ToType (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier) s.pt)) :
            (a : c.pt) → ToType s.pt)
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) : (a : ToType (F.obj j)) → c.pt) x' : c.pt) :
        ToType (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier) s.pt)
      (((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
                (hc.desc
                  (Functor.mapCocone
                    (forget PartOrdEmb (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC :=
                      carrier))
                    s)) :
              TypeCat.Fun c.pt
                (ToType (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier) s.pt)) :
            (a : c.pt) → ToType s.pt)
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) : (a : ToType (F.obj j)) → c.pt) y' : c.pt) :
        ToType (FC := fun (x1 x2 : PartOrdEmb.{u}) ↦ (↑x1 : Type u) ↪o (↑x2 : Type u)) (CC := carrier) s.pt) ↔
    LE.le (α := CoconePt hc)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) : (a : ToType (F.obj j)) → c.pt) x' : c.pt)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) : (a : ToType (F.obj j)) → c.pt) y' : c.pt)
have rw : x' ≤ y' ↔
  LE.le (α := CoconePt hc)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) : (a : ToType (F.obj j)) → c.pt) x' : c.pt)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (c.ι.app j) : (a : ToType (F.obj j)) → c.pt) y' : c.pt)
Tactic: rw [hx, hy, OrderEmbedding.le_iff_le]