Theorem: IsPreconnected.subset_connectedComponentIn
File: Mathlib/Topology/Connected/Basic.lean

Proof steps:
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s t u v : Set α
x : α
F : Set α
hs : IsPreconnected s
hxs : x ∈ s
hsF : s ⊆ F
⊢ s ⊆ connectedComponentIn F x
state_after: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s t u v : Set α
x : α
F : Set α
hs : IsPreconnected s
hxs : x ∈ s
hsF : s ⊆ F
this : IsPreconnected (Subtype.val ⁻¹' s)
⊢ s ⊆ connectedComponentIn F x
tactic: have : IsPreconnected (((↑) : F → α) ⁻¹' s) := by
    refine inducing_subtype_val.isPreconnected_image.mp ?_
    rwa [Subtype.image_preimage_coe, inter_eq_right.mpr hsF]
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s t u v : Set α
x : α
F : Set α
hs : IsPreconnected s
hxs : x ∈ s
hsF : s ⊆ F
this : IsPreconnected (Subtype.val ⁻¹' s)
⊢ s ⊆ connectedComponentIn F x
state_after: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s t u v : Set α
x : α
F : Set α
hs : IsPreconnected s
hxs : x ∈ s
hsF : s ⊆ F
this : IsPreconnected (Subtype.val ⁻¹' s)
h2xs : ⟨x, ⋯⟩ ∈ Subtype.val ⁻¹' s
⊢ s ⊆ connectedComponentIn F x
tactic: have h2xs : (⟨x, hsF hxs⟩ : F) ∈ (↑) ⁻¹' s := by
    rw [mem_preimage]
    exact hxs
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s t u v : Set α
x : α
F : Set α
hs : IsPreconnected s
hxs : x ∈ s
hsF : s ⊆ F
this : IsPreconnected (Subtype.val ⁻¹' s)
h2xs : ⟨x, ⋯⟩ ∈ Subtype.val ⁻¹' s
⊢ s ⊆ connectedComponentIn F x
state_after: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s t u v : Set α
x : α
F : Set α
hs : IsPreconnected s
hxs : x ∈ s
hsF : s ⊆ F
this✝ : IsPreconnected (Subtype.val ⁻¹' s)
h2xs : ⟨x, ⋯⟩ ∈ Subtype.val ⁻¹' s
this : Subtype.val ⁻¹' s ⊆ connectedComponent ⟨x, ⋯⟩
⊢ s ⊆ connectedComponentIn F x
tactic: have := this.subset_connectedComponent h2xs
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s t u v : Set α
x : α
F : Set α
hs : IsPreconnected s
hxs : x ∈ s
hsF : s ⊆ F
this✝ : IsPreconnected (Subtype.val ⁻¹' s)
h2xs : ⟨x, ⋯⟩ ∈ Subtype.val ⁻¹' s
this : Subtype.val ⁻¹' s ⊆ connectedComponent ⟨x, ⋯⟩
⊢ s ⊆ connectedComponentIn F x
state_after: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s t u v : Set α
x : α
F : Set α
hs : IsPreconnected s
hxs : x ∈ s
hsF : s ⊆ F
this✝ : IsPreconnected (Subtype.val ⁻¹' s)
h2xs : ⟨x, ⋯⟩ ∈ Subtype.val ⁻¹' s
this : Subtype.val ⁻¹' s ⊆ connectedComponent ⟨x, ⋯⟩
⊢ s ⊆ Subtype.val '' connectedComponent ⟨x, ⋯⟩
tactic: rw [connectedComponentIn_eq_image (hsF hxs)]
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s t u v : Set α
x : α
F : Set α
hs : IsPreconnected s
hxs : x ∈ s
hsF : s ⊆ F
this✝ : IsPreconnected (Subtype.val ⁻¹' s)
h2xs : ⟨x, ⋯⟩ ∈ Subtype.val ⁻¹' s
this : Subtype.val ⁻¹' s ⊆ connectedComponent ⟨x, ⋯⟩
⊢ s ⊆ Subtype.val '' connectedComponent ⟨x, ⋯⟩
state_after: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s t u v : Set α
x : α
F : Set α
hs : IsPreconnected s
hxs : x ∈ s
hsF : s ⊆ F
this✝ : IsPreconnected (Subtype.val ⁻¹' s)
h2xs : ⟨x, ⋯⟩ ∈ Subtype.val ⁻¹' s
this : Subtype.val ⁻¹' s ⊆ connectedComponent ⟨x, ⋯⟩
⊢ s ⊆ Subtype.val '' (Subtype.val ⁻¹' s)
tactic: refine Subset.trans ?_ (image_subset _ this)
state_before: α : Type u
β : Type v
ι : Type u_1
π : ι → Type u_2
inst✝ : TopologicalSpace α
s t u v : Set α
x : α
F : Set α
hs : IsPreconnected s
hxs : x ∈ s
hsF : s ⊆ F
this✝ : IsPreconnected (Subtype.val ⁻¹' s)
h2xs : ⟨x, ⋯⟩ ∈ Subtype.val ⁻¹' s
this : Subtype.val ⁻¹' s ⊆ connectedComponent ⟨x, ⋯⟩
⊢ s ⊆ Subtype.val '' (Subtype.val ⁻¹' s)
state_after: no goals
tactic: rw [Subtype.image_preimage_coe, inter_eq_right.mpr hsF]