/-- The Gromov-Hausdorff distance between two spaces is bounded by the Hausdorff distance
of isometric copies of the spaces, in any metric space. -/
theorem ghDist_le_hausdorffDist {X : Type u} [MetricSpace X] [CompactSpace X] [Nonempty X]
    {Y : Type v} [MetricSpace Y] [CompactSpace Y] [Nonempty Y] {γ : Type w} [MetricSpace γ]
    {Φ : X → γ} {Ψ : Y → γ} (ha : Isometry Φ) (hb : Isometry Ψ) :
    ghDist X Y ≤ hausdorffDist (range Φ) (range Ψ) := by
  /- For the proof, we want to embed `γ` in `ℓ^∞(ℝ)`, to say that the Hausdorff distance is realized
    in `ℓ^∞(ℝ)` and therefore bounded below by the Gromov-Hausdorff-distance. However, `γ` is not
    separable in general. We restrict to the union of the images of `X` and `Y` in `γ`, which is
    separable and therefore embeddable in `ℓ^∞(ℝ)`. -/
  rcases exists_mem_of_nonempty X with ⟨xX, _⟩
  let s : Set γ := range Φ ∪ range Ψ
  let Φ' : X → Subtype s := fun y => ⟨Φ y, mem_union_left _ (mem_range_self _)⟩
  let Ψ' : Y → Subtype s := fun y => ⟨Ψ y, mem_union_right _ (mem_range_self _)⟩
  have IΦ' : Isometry Φ' := fun x y => ha x y
  have IΨ' : Isometry Ψ' := fun x y => hb x y
  have : IsCompact s := (isCompact_range ha.continuous).union (isCompact_range hb.continuous)
  letI : MetricSpace (Subtype s) := by infer_instance
  haveI : CompactSpace (Subtype s) := ⟨isCompact_iff_isCompact_univ.1 ‹IsCompact s›⟩
  haveI : Nonempty (Subtype s) := ⟨Φ' xX⟩
  have ΦΦ' : Φ = Subtype.val ∘ Φ' := by funext; rfl
  have ΨΨ' : Ψ = Subtype.val ∘ Ψ' := by funext; rfl
  have : hausdorffDist (range Φ) (range Ψ) = hausdorffDist (range Φ') (range Ψ') := by
    rw [ΦΦ', ΨΨ', range_comp, range_comp]
    exact hausdorffDist_image isometry_subtype_coe
  rw [this]
  -- Embed `s` in `ℓ^∞(ℝ)` through its Kuratowski embedding
  let F := kuratowskiEmbedding (Subtype s)
  have : hausdorffDist (F '' range Φ') (F '' range Ψ') = hausdorffDist (range Φ') (range Ψ') :=
    hausdorffDist_image (kuratowskiEmbedding.isometry _)
  rw [← this]
  -- Let `A` and `B` be the images of `X` and `Y` under this embedding. They are in `ℓ^∞(ℝ)`, and
  -- their Hausdorff distance is the same as in the original space.
  let A : NonemptyCompacts ℓ_infty_ℝ :=
    ⟨⟨F '' range Φ',
        (isCompact_range IΦ'.continuous).image (kuratowskiEmbedding.isometry _).continuous⟩,
      (range_nonempty _).image _⟩
  let B : NonemptyCompacts ℓ_infty_ℝ :=
    ⟨⟨F '' range Ψ',
        (isCompact_range IΨ'.continuous).image (kuratowskiEmbedding.isometry _).continuous⟩,
      (range_nonempty _).image _⟩
  have AX : ⟦A⟧ = toGHSpace X := by
    rw [eq_toGHSpace_iff]
    exact ⟨fun x => F (Φ' x), (kuratowskiEmbedding.isometry _).comp IΦ', range_comp _ _⟩
  have BY : ⟦B⟧ = toGHSpace Y := by
    rw [eq_toGHSpace_iff]
    exact ⟨fun x => F (Ψ' x), (kuratowskiEmbedding.isometry _).comp IΨ', range_comp _ _⟩
  refine' csInf_le ⟨0, _⟩ _
  · 

Proof state:
case intro.refine'_1
X : Type u
inst✝⁶ : MetricSpace X
inst✝⁵ : CompactSpace X
inst✝⁴ : Nonempty X
Y : Type v
inst✝³ : MetricSpace Y
inst✝² : CompactSpace Y
inst✝¹ : Nonempty Y
γ : Type w
inst✝ : MetricSpace γ
Φ : X → γ
Ψ : Y → γ
ha : Isometry Φ
hb : Isometry Ψ
xX : X
h✝ : xX ∈ univ
s : Set γ := range Φ ∪ range Ψ
Φ' : X → Subtype s := fun y => { val := Φ y, property := (_ : Φ y ∈ range Φ ∪ range Ψ) }
Ψ' : Y → Subtype s := fun y => { val := Ψ y, property := (_ : Ψ y ∈ range Φ ∪ range Ψ) }
IΦ' : Isometry Φ'
IΨ' : Isometry Ψ'
this✝⁴ : IsCompact s
this✝³ : MetricSpace (Subtype s) := inferInstance
this✝² : CompactSpace (Subtype s)
this✝¹ : Nonempty (Subtype s)
ΦΦ' : Φ = Subtype.val ∘ Φ'
ΨΨ' : Ψ = Subtype.val ∘ Ψ'
this✝ : hausdorffDist (range Φ) (range Ψ) = hausdorffDist (range Φ') (range Ψ')
F : Subtype s → ↥(lp (fun i => ℝ) ⊤) := kuratowskiEmbedding (Subtype s)
this : hausdorffDist (F '' range Φ') (F '' range Ψ') = hausdorffDist (range Φ') (range Ψ')
A : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤) :=
  { toCompacts := { carrier := F '' range Φ', isCompact' := (_ : IsCompact (F '' range Φ')) },
    nonempty' := (_ : Set.Nonempty ((fun a => F a) '' range Φ')) }
B : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤) :=
  { toCompacts := { carrier := F '' range Ψ', isCompact' := (_ : IsCompact (F '' range Ψ')) },
    nonempty' := (_ : Set.Nonempty ((fun a => F a) '' range Ψ')) }
AX : ⟦A⟧ = toGHSpace X
BY : ⟦B⟧ = toGHSpace Y
⊢ 0 ∈ lowerBounds ((fun p => hausdorffDist ↑p.1 ↑p.2) '' {a | ⟦a⟧ = toGHSpace X} ×ˢ {b | ⟦b⟧ = toGHSpace Y})

Next tactic:
simp only [lowerBounds, mem_image, mem_prod, mem_setOf_eq, Prod.exists, and_imp,
      forall_exists_index]