/-- The preimage of a connected component is preconnected if the function has connected fibers
and a subset is closed iff the preimage is. -/
theorem preimage_connectedComponent_connected [TopologicalSpace β] {f : α → β}
    (connected_fibers : ∀ t : β, IsConnected (f ⁻¹' {t}))
    (hcl : ∀ T : Set β, IsClosed T ↔ IsClosed (f ⁻¹' T)) (t : β) :
    IsConnected (f ⁻¹' connectedComponent t) := by
  -- The following proof is essentially https://stacks.math.columbia.edu/tag/0377
  -- although the statement is slightly different
  have hf : Surjective f := Surjective.of_comp fun t : β => (connected_fibers t).1
  refine ⟨Nonempty.preimage connectedComponent_nonempty hf, ?_⟩
  have hT : IsClosed (f ⁻¹' connectedComponent t) :=
    (hcl (connectedComponent t)).1 isClosed_connectedComponent
  -- To show it's preconnected we decompose (f ⁻¹' connectedComponent t) as a subset of two
  -- closed disjoint sets in α. We want to show that it's a subset of either.
  rw [isPreconnected_iff_subset_of_fully_disjoint_closed hT]
  intro u v hu hv huv uv_disj
  -- To do this we decompose connectedComponent t into T₁ and T₂
  -- we will show that connectedComponent t is a subset of either and hence
  -- (f ⁻¹' connectedComponent t) is a subset of u or v
  let T₁ := { t' ∈ connectedComponent t | f ⁻¹' {t'} ⊆ u }
  let T₂ := { t' ∈ connectedComponent t | f ⁻¹' {t'} ⊆ v }
  have fiber_decomp : ∀ t' ∈ connectedComponent t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v := by
    intro t' ht'
    apply isPreconnected_iff_subset_of_disjoint_closed.1 (connected_fibers t').2 u v hu hv
    · exact Subset.trans (preimage_mono (singleton_subset_iff.2 ht')) huv
    rw [uv_disj.inter_eq, inter_empty]
  have T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u := by
    apply eq_of_subset_of_subset
    · rw [← biUnion_preimage_singleton]
      refine' iUnion₂_subset fun t' ht' => subset_inter _ ht'.2
      rw [hf.preimage_subset_preimage_iff, singleton_subset_iff]
      exact ht'.1
    rintro a ⟨hat, hau⟩
    constructor
    · exact mem_preimage.1 hat
    refine (fiber_decomp (f a) (mem_preimage.1 hat)).resolve_right fun h => ?_
    exact uv_disj.subset_compl_right hau (h rfl)
  -- This proof is exactly the same as the above (modulo some symmetry)
  have T₂_v : f ⁻¹' T₂ = f ⁻¹' connectedComponent t ∩ v := by
    apply eq_of_subset_of_subset
    · rw [← biUnion_preimage_singleton]
      refine' iUnion₂_subset fun t' ht' => subset_inter _ ht'.2
      rw [hf.preimage_subset_preimage_iff, singleton_subset_iff]
      exact ht'.1
    rintro a ⟨hat, hav⟩
    constructor
    · exact mem_preimage.1 hat
    · refine (fiber_decomp (f a) (mem_preimage.1 hat)).resolve_left fun h => ?_
      exact uv_disj.subset_compl_left hav (h rfl)
  -- Now we show T₁, T₂ are closed, cover connectedComponent t and are disjoint.
  have hT₁ : IsClosed T₁ := (hcl T₁).2 (T₁_u.symm ▸ IsClosed.inter hT hu)
  have hT₂ : IsClosed T₂ := (hcl T₂).2 (T₂_v.symm ▸ IsClosed.inter hT hv)
  have T_decomp : connectedComponent t ⊆ T₁ ∪ T₂ := fun t' ht' => by
    rw [mem_union t' T₁ T₂]
    cases' fiber_decomp t' ht' with htu htv
    · left
      exact ⟨ht', htu⟩
    right
    exact ⟨ht', htv⟩
  have T_disjoint : Disjoint T₁ T₂ := by
    refine' Disjoint.of_preimage hf _
    rw [T₁_u, T₂_v, disjoint_iff_inter_eq_empty, ← inter_inter_distrib_left, uv_disj.inter_eq,
      inter_empty]
  -- Now we do cases on whether (connectedComponent t) is a subset of T₁ or T₂ to show
  -- that the preimage is a subset of u or v.
  cases' (isPreconnected_iff_subset_of_fully_disjoint_closed isClosed_connectedComponent).1
    isPreconnected_connectedComponent T₁ T₂ hT₁ hT₂ T_decomp T_disjoint with h h
  · left
    rw [Subset.antisymm_iff] at T₁_u
    suffices f ⁻¹' connectedComponent t ⊆ f ⁻¹' T₁
      from (this.trans T₁_u.1).trans (inter_subset_right _ _)
    exact preimage_mono h
  · right
    rw [Subset.antisymm_iff] at T₂_v
    suffices f ⁻¹' connectedComponent t ⊆ f ⁻¹' T₂
      from (this.trans T₂_v.1).trans (inter_subset_right _ _)
    

Proof state:
case inr.h
α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝¹ : TopologicalSpace α
s t✝ u✝ v✝ : Set α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed v
huv : f ⁻¹' connectedComponent t ⊆ u ∪ v
uv_disj : Disjoint u v
T₁ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ u}
T₂ : Set β := {t' | t' ∈ connectedComponent t ∧ f ⁻¹' {t'} ⊆ v}
fiber_decomp : ∀ t' ∈ connectedComponent t, f ⁻¹' {t'} ⊆ u ∨ f ⁻¹' {t'} ⊆ v
T₁_u : f ⁻¹' T₁ = f ⁻¹' connectedComponent t ∩ u
T₂_v : f ⁻¹' T₂ ⊆ f ⁻¹' connectedComponent t ∩ v ∧ f ⁻¹' connectedComponent t ∩ v ⊆ f ⁻¹' T₂
hT₁ : IsClosed T₁
hT₂ : IsClosed T₂
T_decomp : connectedComponent t ⊆ T₁ ∪ T₂
T_disjoint : Disjoint T₁ T₂
h : connectedComponent t ⊆ T₂
⊢ f ⁻¹' connectedComponent t ⊆ f ⁻¹' T₂

Next tactic:
exact preimage_mono h