Theorem: image_subset_closure_compl_image_compl_of_isOpen
File: Mathlib/Topology/ExtremallyDisconnected.lean

tactic: by_cases G_empty : G = ∅
state_before: X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
⊢ ρ '' G ⊆ closure (ρ '' Gᶜ)ᶜ
state_after: case pos
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : G = ∅
⊢ ρ '' G ⊆ closure (ρ '' Gᶜ)ᶜ

case neg
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
⊢ ρ '' G ⊆ closure (ρ '' Gᶜ)ᶜ

tactic: simpa only [G_empty, image_empty] using empty_subset _
state_before: case pos
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : G = ∅
⊢ ρ '' G ⊆ closure (ρ '' Gᶜ)ᶜ
state_after: no goals

tactic: intro a ha
state_before: case neg
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
⊢ ρ '' G ⊆ closure (ρ '' Gᶜ)ᶜ
state_after: case neg
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
a : A
ha : a ∈ ρ '' G
⊢ a ∈ closure (ρ '' Gᶜ)ᶜ

tactic: rw [mem_closure_iff]
state_before: case neg
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
a : A
ha : a ∈ ρ '' G
⊢ a ∈ closure (ρ '' Gᶜ)ᶜ
state_after: case neg
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
a : A
ha : a ∈ ρ '' G
⊢ ∀ (o : Set A), IsOpen o → a ∈ o → (o ∩ (ρ '' Gᶜ)ᶜ).Nonempty

tactic: intro N N_open hN
state_before: case neg
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
a : A
ha : a ∈ ρ '' G
⊢ ∀ (o : Set A), IsOpen o → a ∈ o → (o ∩ (ρ '' Gᶜ)ᶜ).Nonempty
state_after: case neg
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
a : A
ha : a ∈ ρ '' G
N : Set A
N_open : IsOpen N
hN : a ∈ N
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty

tactic: rcases (G.mem_image ρ a).mp ha with ⟨e, he, rfl⟩
state_before: case neg
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
a : A
ha : a ∈ ρ '' G
N : Set A
N_open : IsOpen N
hN : a ∈ N
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty
state_after: case neg.intro.intro
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty

tactic: have nonempty : (G ∩ ρ⁻¹' N).Nonempty := ⟨e, mem_inter he <| mem_preimage.mpr hN⟩
state_before: case neg.intro.intro
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty
state_after: case neg.intro.intro
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
nonempty : (G ∩ ρ ⁻¹' N).Nonempty
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty

tactic: have is_open : IsOpen <| G ∩ ρ⁻¹' N := hG.inter <| N_open.preimage ρ_cont
state_before: case neg.intro.intro
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
nonempty : (G ∩ ρ ⁻¹' N).Nonempty
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty
state_after: case neg.intro.intro
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
nonempty : (G ∩ ρ ⁻¹' N).Nonempty
is_open : IsOpen (G ∩ ρ ⁻¹' N)
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty

tactic: have ne_univ : ρ '' (G ∩ ρ⁻¹' N)ᶜ ≠ univ :=
  zorn_subset _ (compl_ne_univ.mpr nonempty) is_open.isClosed_compl
state_before: case neg.intro.intro
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
nonempty : (G ∩ ρ ⁻¹' N).Nonempty
is_open : IsOpen (G ∩ ρ ⁻¹' N)
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty
state_after: case neg.intro.intro
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
nonempty : (G ∩ ρ ⁻¹' N).Nonempty
is_open : IsOpen (G ∩ ρ ⁻¹' N)
ne_univ : ρ '' (G ∩ ρ ⁻¹' N)ᶜ ≠ univ
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty

tactic: rcases nonempty_compl.mpr ne_univ with ⟨x, hx⟩
state_before: case neg.intro.intro
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
nonempty : (G ∩ ρ ⁻¹' N).Nonempty
is_open : IsOpen (G ∩ ρ ⁻¹' N)
ne_univ : ρ '' (G ∩ ρ ⁻¹' N)ᶜ ≠ univ
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty
state_after: case neg.intro.intro.intro
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
nonempty : (G ∩ ρ ⁻¹' N).Nonempty
is_open : IsOpen (G ∩ ρ ⁻¹' N)
ne_univ : ρ '' (G ∩ ρ ⁻¹' N)ᶜ ≠ univ
x : A
hx : x ∈ (ρ '' (G ∩ ρ ⁻¹' N)ᶜ)ᶜ
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty

tactic: have hx' : x ∈ (ρ '' Gᶜ)ᶜ := fun h => hx <| image_subset ρ (by simp) h
state_before: case neg.intro.intro.intro
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
nonempty : (G ∩ ρ ⁻¹' N).Nonempty
is_open : IsOpen (G ∩ ρ ⁻¹' N)
ne_univ : ρ '' (G ∩ ρ ⁻¹' N)ᶜ ≠ univ
x : A
hx : x ∈ (ρ '' (G ∩ ρ ⁻¹' N)ᶜ)ᶜ
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty
state_after: case neg.intro.intro.intro
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
nonempty : (G ∩ ρ ⁻¹' N).Nonempty
is_open : IsOpen (G ∩ ρ ⁻¹' N)
ne_univ : ρ '' (G ∩ ρ ⁻¹' N)ᶜ ≠ univ
x : A
hx : x ∈ (ρ '' (G ∩ ρ ⁻¹' N)ᶜ)ᶜ
hx' : x ∈ (ρ '' Gᶜ)ᶜ
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty

tactic: rcases ρ_surj x with ⟨y, rfl⟩
state_before: case neg.intro.intro.intro
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
nonempty : (G ∩ ρ ⁻¹' N).Nonempty
is_open : IsOpen (G ∩ ρ ⁻¹' N)
ne_univ : ρ '' (G ∩ ρ ⁻¹' N)ᶜ ≠ univ
x : A
hx : x ∈ (ρ '' (G ∩ ρ ⁻¹' N)ᶜ)ᶜ
hx' : x ∈ (ρ '' Gᶜ)ᶜ
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty
state_after: case neg.intro.intro.intro.intro
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
nonempty : (G ∩ ρ ⁻¹' N).Nonempty
is_open : IsOpen (G ∩ ρ ⁻¹' N)
ne_univ : ρ '' (G ∩ ρ ⁻¹' N)ᶜ ≠ univ
y : E
hx : ρ y ∈ (ρ '' (G ∩ ρ ⁻¹' N)ᶜ)ᶜ
hx' : ρ y ∈ (ρ '' Gᶜ)ᶜ
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty

tactic: have hy : y ∈ G ∩ ρ⁻¹' N := by simpa using mt (mem_image_of_mem ρ) <| mem_compl hx
state_before: case neg.intro.intro.intro.intro
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
nonempty : (G ∩ ρ ⁻¹' N).Nonempty
is_open : IsOpen (G ∩ ρ ⁻¹' N)
ne_univ : ρ '' (G ∩ ρ ⁻¹' N)ᶜ ≠ univ
y : E
hx : ρ y ∈ (ρ '' (G ∩ ρ ⁻¹' N)ᶜ)ᶜ
hx' : ρ y ∈ (ρ '' Gᶜ)ᶜ
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty
state_after: case neg.intro.intro.intro.intro
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
nonempty : (G ∩ ρ ⁻¹' N).Nonempty
is_open : IsOpen (G ∩ ρ ⁻¹' N)
ne_univ : ρ '' (G ∩ ρ ⁻¹' N)ᶜ ≠ univ
y : E
hx : ρ y ∈ (ρ '' (G ∩ ρ ⁻¹' N)ᶜ)ᶜ
hx' : ρ y ∈ (ρ '' Gᶜ)ᶜ
hy : y ∈ G ∩ ρ ⁻¹' N
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty

tactic: exact ⟨ρ y, mem_inter (mem_preimage.mp <| mem_of_mem_inter_right hy) hx'⟩
state_before: case neg.intro.intro.intro.intro
X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
nonempty : (G ∩ ρ ⁻¹' N).Nonempty
is_open : IsOpen (G ∩ ρ ⁻¹' N)
ne_univ : ρ '' (G ∩ ρ ⁻¹' N)ᶜ ≠ univ
y : E
hx : ρ y ∈ (ρ '' (G ∩ ρ ⁻¹' N)ᶜ)ᶜ
hx' : ρ y ∈ (ρ '' Gᶜ)ᶜ
hy : y ∈ G ∩ ρ ⁻¹' N
⊢ (N ∩ (ρ '' Gᶜ)ᶜ).Nonempty
state_after: no goals

tactic: simp
state_before: X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
nonempty : (G ∩ ρ ⁻¹' N).Nonempty
is_open : IsOpen (G ∩ ρ ⁻¹' N)
ne_univ : ρ '' (G ∩ ρ ⁻¹' N)ᶜ ≠ univ
x : A
hx : x ∈ (ρ '' (G ∩ ρ ⁻¹' N)ᶜ)ᶜ
h : x ∈ ρ '' Gᶜ
⊢ Gᶜ ⊆ (G ∩ ρ ⁻¹' N)ᶜ
state_after: no goals

tactic: simpa using mt (mem_image_of_mem ρ) <| mem_compl hx
state_before: X : Type u
inst✝³ : TopologicalSpace X
A D E : Type u
inst✝² : TopologicalSpace A
inst✝¹ : TopologicalSpace D
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN : ρ e ∈ N
nonempty : (G ∩ ρ ⁻¹' N).Nonempty
is_open : IsOpen (G ∩ ρ ⁻¹' N)
ne_univ : ρ '' (G ∩ ρ ⁻¹' N)ᶜ ≠ univ
y : E
hx : ρ y ∈ (ρ '' (G ∩ ρ ⁻¹' N)ᶜ)ᶜ
hx' : ρ y ∈ (ρ '' Gᶜ)ᶜ
⊢ y ∈ G ∩ ρ ⁻¹' N
state_after: no goals