Theorem: OpenEmbedding.quasiSober
File: Mathlib/Topology/Sober.lean

tactic: have hS'' := hS.image f hf.continuous.continuousOn
state_before: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
S✝ : Set α
hS : IsIrreducible S✝
hS' : IsClosed S✝
⊢ ∃ x, IsGenericPoint x S✝

tactic: obtain ⟨x, hx⟩ := QuasiSober.sober hS''.closure isClosed_closure
state_before: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
S✝ : Set α
hS : IsIrreducible S✝
hS' : IsClosed S✝
hS'' : IsIrreducible (f '' S✝)
⊢ ∃ x, IsGenericPoint x S✝

tactic: obtain ⟨T, hT, rfl⟩ := hf.toInducing.isClosed_iff.mp hS'
state_before: case intro
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
S✝ : Set α
hS : IsIrreducible S✝
hS' : IsClosed S✝
hS'' : IsIrreducible (f '' S✝)
x : β
hx : IsGenericPoint x (closure (f '' S✝))
⊢ ∃ x, IsGenericPoint x S✝

tactic: rw [image_preimage_eq_inter_range] at hx hS''
state_before: case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
x : β
T : Set β
hT : IsClosed T
hS : IsIrreducible (f ⁻¹' T)
hS' : IsClosed (f ⁻¹' T)
hS'' : IsIrreducible (f '' (f ⁻¹' T))
hx : IsGenericPoint x (closure (f '' (f ⁻¹' T)))
⊢ ∃ x, IsGenericPoint x (f ⁻¹' T)

tactic: have hxT : x ∈ T := by
  rw [← hT.closure_eq]
  exact closure_mono (inter_subset_left _ _) hx.mem
state_before: case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
x : β
T : Set β
hT : IsClosed T
hS : IsIrreducible (f ⁻¹' T)
hS' : IsClosed (f ⁻¹' T)
hS'' : IsIrreducible (T ∩ range f)
hx : IsGenericPoint x (closure (T ∩ range f))
⊢ ∃ x, IsGenericPoint x (f ⁻¹' T)

tactic: obtain ⟨y, rfl⟩ : x ∈ range f := by
  rw [hx.mem_open_set_iff hf.open_range]
  refine' Nonempty.mono _ hS''.1
  simpa using subset_closure
state_before: case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
x : β
T : Set β
hT : IsClosed T
hS : IsIrreducible (f ⁻¹' T)
hS' : IsClosed (f ⁻¹' T)
hS'' : IsIrreducible (T ∩ range f)
hx : IsGenericPoint x (closure (T ∩ range f))
hxT : x ∈ T
⊢ ∃ x, IsGenericPoint x (f ⁻¹' T)

tactic: use y
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
T : Set β
hT : IsClosed T
hS : IsIrreducible (f ⁻¹' T)
hS' : IsClosed (f ⁻¹' T)
hS'' : IsIrreducible (T ∩ range f)
y : α
hx : IsGenericPoint (f y) (closure (T ∩ range f))
hxT : f y ∈ T
⊢ ∃ x, IsGenericPoint x (f ⁻¹' T)

tactic: change _ = _
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
T : Set β
hT : IsClosed T
hS : IsIrreducible (f ⁻¹' T)
hS' : IsClosed (f ⁻¹' T)
hS'' : IsIrreducible (T ∩ range f)
y : α
hx : IsGenericPoint (f y) (closure (T ∩ range f))
hxT : f y ∈ T
⊢ IsGenericPoint y (f ⁻¹' T)

tactic: rw [hf.toEmbedding.closure_eq_preimage_closure_image, image_singleton, show _ = _ from hx]
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
T : Set β
hT : IsClosed T
hS : IsIrreducible (f ⁻¹' T)
hS' : IsClosed (f ⁻¹' T)
hS'' : IsIrreducible (T ∩ range f)
y : α
hx : IsGenericPoint (f y) (closure (T ∩ range f))
hxT : f y ∈ T
⊢ closure {y} = f ⁻¹' T

tactic: apply image_injective.mpr hf.inj
state_before: case intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
T : Set β
hT : IsClosed T
hS : IsIrreducible (f ⁻¹' T)
hS' : IsClosed (f ⁻¹' T)
hS'' : IsIrreducible (T ∩ range f)
y : α
hx : IsGenericPoint (f y) (closure (T ∩ range f))
hxT : f y ∈ T
⊢ f ⁻¹' closure (T ∩ range f) = f ⁻¹' T

tactic: ext z
state_before: case intro.intro.intro.intro.a
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
T : Set β
hT : IsClosed T
hS : IsIrreducible (f ⁻¹' T)
hS' : IsClosed (f ⁻¹' T)
hS'' : IsIrreducible (T ∩ range f)
y : α
hx : IsGenericPoint (f y) (closure (T ∩ range f))
hxT : f y ∈ T
⊢ f '' (f ⁻¹' closure (T ∩ range f)) = f '' (f ⁻¹' T)

tactic: simp only [image_preimage_eq_inter_range, mem_inter_iff, and_congr_left_iff]
state_before: case intro.intro.intro.intro.a.h
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
T : Set β
hT : IsClosed T
hS : IsIrreducible (f ⁻¹' T)
hS' : IsClosed (f ⁻¹' T)
hS'' : IsIrreducible (T ∩ range f)
y : α
hx : IsGenericPoint (f y) (closure (T ∩ range f))
hxT : f y ∈ T
z : β
⊢ z ∈ f '' (f ⁻¹' closure (T ∩ range f)) ↔ z ∈ f '' (f ⁻¹' T)

tactic: exact fun hy => ⟨fun h => hT.closure_eq ▸ closure_mono (inter_subset_left _ _) h,
  fun h => subset_closure ⟨h, hy⟩⟩
state_before: case intro.intro.intro.intro.a.h
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
T : Set β
hT : IsClosed T
hS : IsIrreducible (f ⁻¹' T)
hS' : IsClosed (f ⁻¹' T)
hS'' : IsIrreducible (T ∩ range f)
y : α
hx : IsGenericPoint (f y) (closure (T ∩ range f))
hxT : f y ∈ T
z : β
⊢ z ∈ range f → (z ∈ closure (T ∩ range f) ↔ z ∈ T)

tactic: rw [← hT.closure_eq]
state_before: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
x : β
T : Set β
hT : IsClosed T
hS : IsIrreducible (f ⁻¹' T)
hS' : IsClosed (f ⁻¹' T)
hS'' : IsIrreducible (T ∩ range f)
hx : IsGenericPoint x (closure (T ∩ range f))
⊢ x ∈ T

tactic: exact closure_mono (inter_subset_left _ _) hx.mem
state_before: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
x : β
T : Set β
hT : IsClosed T
hS : IsIrreducible (f ⁻¹' T)
hS' : IsClosed (f ⁻¹' T)
hS'' : IsIrreducible (T ∩ range f)
hx : IsGenericPoint x (closure (T ∩ range f))
⊢ x ∈ closure T

tactic: rw [hx.mem_open_set_iff hf.open_range]
state_before: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
x : β
T : Set β
hT : IsClosed T
hS : IsIrreducible (f ⁻¹' T)
hS' : IsClosed (f ⁻¹' T)
hS'' : IsIrreducible (T ∩ range f)
hx : IsGenericPoint x (closure (T ∩ range f))
hxT : x ∈ T
⊢ x ∈ range f

tactic: refine' Nonempty.mono _ hS''.1
state_before: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
x : β
T : Set β
hT : IsClosed T
hS : IsIrreducible (f ⁻¹' T)
hS' : IsClosed (f ⁻¹' T)
hS'' : IsIrreducible (T ∩ range f)
hx : IsGenericPoint x (closure (T ∩ range f))
hxT : x ∈ T
⊢ Set.Nonempty (closure (T ∩ range f) ∩ range f)

tactic: simpa using subset_closure
state_before: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
hf : OpenEmbedding f
inst✝ : QuasiSober β
x : β
T : Set β
hT : IsClosed T
hS : IsIrreducible (f ⁻¹' T)
hS' : IsClosed (f ⁻¹' T)
hS'' : IsIrreducible (T ∩ range f)
hx : IsGenericPoint x (closure (T ∩ range f))
hxT : x ∈ T
⊢ T ∩ range f ⊆ closure (T ∩ range f) ∩ range f