-- Mathlib/Topology/CompactOpen.lean
theorem continuous_comp' : Continuous fun x : C(X, Y) × C(Y, Z) => x.2.comp x.1

Proof state:
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : LocallyCompactPair Y Z
⊢ ∀ (x : C(X, Y) × C(Y, Z)) (K : Set X),
    IsCompact K →
      ∀ (U : Set Z),
        IsOpen U → MapsTo (⇑(x.2.comp x.1)) K U → ∀ᶠ (a : C(X, Y) × C(Y, Z)) in 𝓝 x, MapsTo (⇑(a.2.comp a.1)) K U

Tactic:
intro ⟨f, g⟩ K hK U hU (hKU : MapsTo (g ∘ f) K U)

Result:
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : LocallyCompactPair Y Z
f : C(X, Y)
g : C(Y, Z)
K : Set X
hK : IsCompact K
U : Set Z
hU : IsOpen U
hKU : MapsTo (⇑g ∘ ⇑f) K U
⊢ ∀ᶠ (a : C(X, Y) × C(Y, Z)) in 𝓝 (f, g), MapsTo (⇑(a.2.comp a.1)) K U

Proof state:
case intro.intro.intro
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : LocallyCompactPair Y Z
f : C(X, Y)
g : C(Y, Z)
K : Set X
hK : IsCompact K
U : Set Z
hU : IsOpen U
hKU : MapsTo (⇑g ∘ ⇑f) K U
L : Set Y
hKL : L ∈ 𝓝ˢ (⇑f '' K)
hLc : IsCompact L
hLU : MapsTo (⇑g) L U
⊢ ∀ᶠ (a : C(X, Y) × C(Y, Z)) in 𝓝 (f, g), MapsTo (⇑(a.2.comp a.1)) K U

Tactic:
rw [← subset_interior_iff_mem_nhdsSet, ← mapsTo'] at hKL

Result:
case intro.intro.intro
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : LocallyCompactPair Y Z
f : C(X, Y)
g : C(Y, Z)
K : Set X
hK : IsCompact K
U : Set Z
hU : IsOpen U
hKU : MapsTo (⇑g ∘ ⇑f) K U
L : Set Y
hKL : MapsTo (⇑f) K (interior L)
hLc : IsCompact L
hLU : MapsTo (⇑g) L U
⊢ ∀ᶠ (a : C(X, Y) × C(Y, Z)) in 𝓝 (f, g), MapsTo (⇑(a.2.comp a.1)) K U

Proof state:
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : LocallyCompactPair Y Z
f : C(X, Y)
g : C(Y, Z)
K : Set X
hK : IsCompact K
U : Set Z
hU : IsOpen U
hKU : MapsTo (⇑g ∘ ⇑f) K U
⊢ ∀ᶠ (a : C(X, Y) × C(Y, Z)) in 𝓝 (f, g), MapsTo (⇑(a.2.comp a.1)) K U

Tactic:
obtain ⟨L, hKL, hLc, hLU⟩ : ∃ L ∈ 𝓝ˢ (f '' K), IsCompact L ∧ MapsTo g L U :=
  exists_mem_nhdsSet_isCompact_mapsTo g.continuous (hK.image f.continuous) hU
    (mapsTo_image_iff.2 hKU)

Result:
case intro.intro.intro
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : LocallyCompactPair Y Z
f : C(X, Y)
g : C(Y, Z)
K : Set X
hK : IsCompact K
U : Set Z
hU : IsOpen U
hKU : MapsTo (⇑g ∘ ⇑f) K U
L : Set Y
hKL : L ∈ 𝓝ˢ (⇑f '' K)
hLc : IsCompact L
hLU : MapsTo (⇑g) L U
⊢ ∀ᶠ (a : C(X, Y) × C(Y, Z)) in 𝓝 (f, g), MapsTo (⇑(a.2.comp a.1)) K U

Proof state:
case intro.intro.intro
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : LocallyCompactPair Y Z
f : C(X, Y)
g : C(Y, Z)
K : Set X
hK : IsCompact K
U : Set Z
hU : IsOpen U
hKU : MapsTo (⇑g ∘ ⇑f) K U
L : Set Y
hKL : MapsTo (⇑f) K (interior L)
hLc : IsCompact L
hLU : MapsTo (⇑g) L U
⊢ ∀ᶠ (a : C(X, Y) × C(Y, Z)) in 𝓝 (f, g), MapsTo (⇑(a.2.comp a.1)) K U

Tactic:
exact ((eventually_mapsTo hK isOpen_interior hKL).prod_nhds
  (eventually_mapsTo hLc hU hLU)).mono fun ⟨f', g'⟩ ⟨hf', hg'⟩ ↦
    hg'.comp <| hf'.mono_right interior_subset

Result:
no goals

Proof state:
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : LocallyCompactPair Y Z
⊢ Continuous fun x => x.2.comp x.1

Tactic:
simp_rw [continuous_iff_continuousAt, ContinuousAt, tendsto_nhds_compactOpen]

Result:
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : LocallyCompactPair Y Z
⊢ ∀ (x : C(X, Y) × C(Y, Z)) (K : Set X),
    IsCompact K →
      ∀ (U : Set Z),
        IsOpen U → MapsTo (⇑(x.2.comp x.1)) K U → ∀ᶠ (a : C(X, Y) × C(Y, Z)) in 𝓝 x, MapsTo (⇑(a.2.comp a.1)) K U