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Result

Found 385 declarations mentioning Filter.comap. Of these, only the first 200 are shown.

• Filter.comap 📋 Mathlib.Order.Filter.Defs
  {α : Type u_1} {β : Type u_2} (m : α → β) (f : Filter β) : Filter α
• Filter.prod_eq_inf 📋 Mathlib.Order.Filter.Defs
  {α : Type u_1} {β : Type u_2} (f : Filter α) (g : Filter β) : f ×ˢ g = Filter.comap Prod.fst f ⊓ Filter.comap Prod.snd g
• Filter.comap_id 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {f : Filter α} : Filter.comap id f = f
• Filter.comap_id' 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {f : Filter α} : Filter.comap (fun x => x) f = f
• Filter.neBot_of_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {g : Filter β} {m : α → β} (h : (Filter.comap m g).NeBot) : g.NeBot
• Filter.comap_injective 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} (hf : Function.Surjective f) : Function.Injective (Filter.comap f)
• Filter.NeBot.comap_of_surj 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter β} {m : α → β} (hf : f.NeBot) (hm : Function.Surjective m) : (Filter.comap m f).NeBot
• Function.Commute.filter_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {f g : α → α} (h : Function.Commute f g) : Function.Commute (Filter.comap f) (Filter.comap g)
• Filter.comap_bot 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {m : α → β} : Filter.comap m ⊥ = ⊥
• Filter.comap_top 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {m : α → β} : Filter.comap m ⊤ = ⊤
• Filter.comap_fst_neBot 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} [Nonempty β] {f : Filter α} [f.NeBot] : (Filter.comap Prod.fst f).NeBot
• Filter.comap_snd_neBot 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} [Nonempty α] {f : Filter β} [f.NeBot] : (Filter.comap Prod.snd f).NeBot
• Filter.comap_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} (h : Function.Injective m) : Filter.comap m (Filter.map m f) = f
• Filter.comap_mono 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {m : α → β} : Monotone (Filter.comap m)
• Filter.comap_principal 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {m : α → β} {t : Set β} : Filter.comap m (Filter.principal t) = Filter.principal (m ⁻¹' t)
• Filter.map_comap_of_surjective 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} (hf : Function.Surjective f) (l : Filter β) : Filter.map f (Filter.comap f l) = l
• Filter.comap_fst_neBot_iff 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} : (Filter.comap Prod.fst f).NeBot ↔ f.NeBot ∧ Nonempty β
• Filter.comap_snd_neBot_iff 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter β} : (Filter.comap Prod.snd f).NeBot ↔ Nonempty α ∧ f.NeBot
• Function.LeftInverse.filter_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} {g : β → α} (hfg : Function.LeftInverse g f) : Function.RightInverse (Filter.comap g) (Filter.comap f)
• Function.RightInverse.filter_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} {g : β → α} (hfg : Function.RightInverse g f) : Function.LeftInverse (Filter.comap g) (Filter.comap f)
• Filter.Eventually.comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {g : Filter β} {p : β → Prop} (hf : ∀ᶠ (b : β) in g, p b) (f : α → β) : ∀ᶠ (a : α) in Filter.comap f g, p (f a)
• Filter.gc_comap_kernMap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (m : α → β) : GaloisConnection (Filter.comap m) (Filter.kernMap m)
• Filter.gc_map_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (m : α → β) : GaloisConnection (Filter.map m) (Filter.comap m)
• Filter.le_comap_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} : f ≤ Filter.comap m (Filter.map m f)
• Filter.le_comap_top 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (f : α → β) (l : Filter α) : l ≤ Filter.comap f ⊤
• Filter.map_comap_le 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {g : Filter β} {m : α → β} : Filter.map m (Filter.comap m g) ≤ g
• Filter.NeBot.comap_of_range_mem 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter β} {m : α → β} : f.NeBot → ∀ (hm : Set.range m ∈ f), (Filter.comap m f).NeBot
• Filter.comap_inl_map_inr 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {g : Filter β} : Filter.comap Sum.inl (Filter.map Sum.inr g) = ⊥
• Filter.comap_inr_map_inl 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} : Filter.comap Sum.inr (Filter.map Sum.inl f) = ⊥
• Filter.comap_neBot_iff_frequently 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter β} {m : α → β} : (Filter.comap m f).NeBot ↔ ∃ᶠ (y : β) in f, y ∈ Set.range m
• Filter.comap_eval_neBot 📋 Mathlib.Order.Filter.Map
  {ι : Type u_6} {α : ι → Type u_7} [∀ (j : ι), Nonempty (α j)] (i : ι) (f : Filter (α i)) [f.NeBot] : (Filter.comap (Function.eval i) f).NeBot
• Filter.comap_inf_principal_range 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {g : Filter β} {m : α → β} : Filter.comap m (g ⊓ Filter.principal (Set.range m)) = Filter.comap m g
• Filter.comap_pure 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {m : α → β} {b : β} : Filter.comap m (pure b) = Filter.principal (m ⁻¹' {b})
• Filter.map_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (f : Filter β) (m : α → β) : Filter.map m (Filter.comap m f) = f ⊓ Filter.principal (Set.range m)
• Filter.comap_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : Filter α} {m : γ → β} {n : β → α} : Filter.comap m (Filter.comap n f) = Filter.comap (n ∘ m) f
• Filter.comap_eval_neBot_iff 📋 Mathlib.Order.Filter.Map
  {ι : Type u_6} {α : ι → Type u_7} [∀ (j : ι), Nonempty (α j)] {i : ι} {f : Filter (α i)} : (Filter.comap (Function.eval i) f).NeBot ↔ f.NeBot
• Filter.eventually_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} {l : Filter β} {p : α → Prop} : (∀ᶠ (a : α) in Filter.comap f l, p a) ↔ ∀ᶠ (b : β) in l, ∀ (a : α), f a = b → p a
• Filter.map_comap_of_mem 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter β} {m : α → β} (hf : Set.range m ∈ f) : Filter.map m (Filter.comap m f) = f
• Filter.NeBot.comap_of_image_mem 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter β} {m : α → β} (hf : f.NeBot) {s : Set α} (hs : m '' s ∈ f) : (Filter.comap m f).NeBot
• Function.Semiconj.filter_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} {ga : α → α} {gb : β → β} (h : Function.Semiconj f ga gb) : Function.Semiconj (Filter.comap f) (Filter.comap gb) (Filter.comap ga)
• Filter.comap_eval_neBot_iff' 📋 Mathlib.Order.Filter.Map
  {ι : Type u_6} {α : ι → Type u_7} {i : ι} {f : Filter (α i)} : (Filter.comap (Function.eval i) f).NeBot ↔ (∀ (j : ι), Nonempty (α j)) ∧ f.NeBot
• Filter.comap_neBot_iff_compl_range 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter β} {m : α → β} : (Filter.comap m f).NeBot ↔ (Set.range m)ᶜ ∉ f
• Filter.comap_surjective_eq_bot 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter β} {m : α → β} (hm : Function.Surjective m) : Filter.comap m f = ⊥ ↔ f = ⊥
• Filter.map_swap_eq_comap_swap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter (α × β)} : Filter.map Prod.swap f = Filter.comap Prod.swap f
• Filter.frequently_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} {l : Filter β} {p : α → Prop} : (∃ᶠ (a : α) in Filter.comap f l, p a) ↔ ∃ᶠ (b : β) in l, ∃ a, f a = b ∧ p a
• Filter.preimage_mem_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {g : Filter β} {m : α → β} {t : Set β} (ht : t ∈ g) : m ⁻¹' t ∈ Filter.comap m g
• Filter.comap_iSup 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {ι : Sort u_6} {f : ι → Filter β} {m : α → β} : Filter.comap m (iSup f) = ⨆ i, Filter.comap m (f i)
• Filter.mem_comap'' 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} {l : Filter β} {s : Set α} : s ∈ Filter.comap f l ↔ Set.kernImage f s ∈ l
• Filter.neBot_inf_comap_iff_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} {F : Filter α} {G : Filter β} : (F ⊓ Filter.comap f G).NeBot ↔ (Filter.map f F ⊓ G).NeBot
• Filter.neBot_inf_comap_iff_map' 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} {F : Filter α} {G : Filter β} : (Filter.comap f G ⊓ F).NeBot ↔ (G ⊓ Filter.map f F).NeBot
• Filter.comap_neBot 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter β} {m : α → β} (hm : ∀ t ∈ f, ∃ a, m a ∈ t) : (Filter.comap m f).NeBot
• Filter.comap_eq_bot_iff_compl_range 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter β} {m : α → β} : Filter.comap m f = ⊥ ↔ (Set.range m)ᶜ ∈ f
• Filter.comap_iInf 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {ι : Sort u_5} {m : α → β} {f : ι → Filter β} : Filter.comap m (⨅ i, f i) = ⨅ i, Filter.comap m (f i)
• Filter.comap_inf 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {g₁ g₂ : Filter β} {m : α → β} : Filter.comap m (g₁ ⊓ g₂) = Filter.comap m g₁ ⊓ Filter.comap m g₂
• Filter.comap_neBot_iff 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter β} {m : α → β} : (Filter.comap m f).NeBot ↔ ∀ t ∈ f, ∃ a, m a ∈ t
• Filter.comap_sup 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {g₁ g₂ : Filter β} {m : α → β} : Filter.comap m (g₁ ⊔ g₂) = Filter.comap m g₁ ⊔ Filter.comap m g₂
• Filter.push_pull 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (f : α → β) (F : Filter α) (G : Filter β) : Filter.map f (F ⊓ Filter.comap f G) = Filter.map f F ⊓ G
• Filter.push_pull' 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (f : α → β) (F : Filter α) (G : Filter β) : Filter.map f (Filter.comap f G ⊓ F) = G ⊓ Filter.map f F
• Filter.EventuallyEq.comp_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {F : Filter β} {f g : β → γ} (h : α → β) (hfg : f =ᶠ[F] g) : f ∘ h =ᶠ[Filter.comap h F] g ∘ h
• Filter.comap_const_of_mem 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {g : Filter β} {x : β} (h : ∀ t ∈ g, x ∈ t) : Filter.comap (fun x_1 => x) g = ⊤
• Filter.comap_const_of_notMem 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {g : Filter β} {t : Set β} {x : β} (ht : t ∈ g) (hx : x ∉ t) : Filter.comap (fun x_1 => x) g = ⊥
• Filter.comap_inf_principal_neBot_of_image_mem 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter β} {m : α → β} (hf : f.NeBot) {s : Set α} (hs : m '' s ∈ f) : (Filter.comap m f ⊓ Filter.principal s).NeBot
• Filter.generate_image_preimage_le_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (U : Set (Set α)) (f : β → α) : Filter.generate ((fun x => f ⁻¹' x) '' U) ≤ Filter.comap f (Filter.generate U)
• Filter.comap_le_iff_le_kernMap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {g : Filter β} {m : α → β} {f : Filter α} : Filter.comap m g ≤ f ↔ g ≤ Filter.kernMap m f
• Filter.map_eq_comap_of_inverse 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} {n : β → α} (h₁ : m ∘ n = id) (h₂ : n ∘ m = id) : Filter.map m f = Filter.comap n f
• Filter.map_le_iff_le_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {g : Filter β} {m : α → β} : Filter.map m f ≤ g ↔ f ≤ Filter.comap m g
• Filter.compl_mem_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} {l : Filter β} {s : Set α} : sᶜ ∈ Filter.comap f l ↔ (f '' s)ᶜ ∈ l
• Filter.mem_comap_iff_compl 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} {l : Filter β} {s : Set α} : s ∈ Filter.comap f l ↔ (f '' sᶜ)ᶜ ∈ l
• Filter.image_mem_of_mem_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {c : β → α} (h : Set.range c ∈ f) {W : Set β} (W_in : W ∈ Filter.comap c f) : c '' W ∈ f
• Filter.mem_comap' 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} {l : Filter β} {s : Set α} : s ∈ Filter.comap f l ↔ {y | ∀ ⦃x : α⦄, f x = y → x ∈ s} ∈ l
• Filter.mem_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {g : Filter β} {m : α → β} {s : Set α} : s ∈ Filter.comap m g ↔ ∃ t ∈ g, m ⁻¹' t ⊆ s
• Filter.sInter_comap_sets 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (f : α → β) (F : Filter β) : ⋂₀ (Filter.comap f F).sets = ⋂ U ∈ F, f ⁻¹' U
• Filter.comap_comm 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {φ : α → β} {θ : α → γ} {ψ : β → δ} {ρ : γ → δ} (H : ψ ∘ φ = ρ ∘ θ) (G : Filter δ) : Filter.comap φ (Filter.comap ψ G) = Filter.comap θ (Filter.comap ρ G)
• Filter.mem_comap_iff 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter β} {m : α → β} (inj : Function.Injective m) (large : Set.range m ∈ f) {S : Set α} : S ∈ Filter.comap m f ↔ m '' S ∈ f
• Filter.comap_equiv_symm 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (e : α ≃ β) (f : Filter α) : Filter.comap (⇑e.symm) f = Filter.map (⇑e) f
• Filter.comap_le_comap_iff 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f g : Filter β} {m : α → β} (hf : Set.range m ∈ f) : Filter.comap m f ≤ Filter.comap m g ↔ f ≤ g
• Filter.map_comap_inl_sup_map_comap_inr 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (l : Filter (α ⊕ β)) : Filter.map Sum.inl (Filter.comap Sum.inl l) ⊔ Filter.map Sum.inr (Filter.comap Sum.inr l) = l
• Filter.map_equiv_symm 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (e : α ≃ β) (f : Filter β) : Filter.map (⇑e.symm) f = Filter.comap (⇑e) f
• Filter.map_sumElim_eq 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} (l : Filter (α ⊕ β)) (m₁ : α → γ) (m₂ : β → γ) : Filter.map (Sum.elim m₁ m₂) l = Filter.map m₁ (Filter.comap Sum.inl l) ⊔ Filter.map m₂ (Filter.comap Sum.inr l)
• Filter.comap_coe_neBot_of_le_principal 📋 Mathlib.Order.Filter.Map
  {γ : Type u_3} {s : Set γ} {l : Filter γ} [h : l.NeBot] (h' : l ≤ Filter.principal s) : (Filter.comap Subtype.val l).NeBot
• Filter.principal_subtype 📋 Mathlib.Order.Filter.Map
  {α : Type u_6} (s : Set α) (t : Set ↑s) : Filter.principal t = Filter.comap Subtype.val (Filter.principal (Subtype.val '' t))
• Filter.comap_sSup 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {s : Set (Filter β)} {m : α → β} : Filter.comap m (sSup s) = ⨆ f ∈ s, Filter.comap m f
• Filter.comap_sumElim_eq 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} (l : Filter γ) (m₁ : α → γ) (m₂ : β → γ) : Filter.comap (Sum.elim m₁ m₂) l = Filter.map Sum.inl (Filter.comap m₁ l) ⊔ Filter.map Sum.inr (Filter.comap m₂ l)
• Filter.disjoint_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {g₁ g₂ : Filter β} {m : α → β} (h : Disjoint g₁ g₂) : Disjoint (Filter.comap m g₁) (Filter.comap m g₂)
• Filter.disjoint_comap_iff_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} {F : Filter α} {G : Filter β} : Disjoint F (Filter.comap f G) ↔ Disjoint (Filter.map f F) G
• Filter.disjoint_comap_iff_map' 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} {F : Filter α} {G : Filter β} : Disjoint (Filter.comap f G) F ↔ Disjoint G (Filter.map f F)
• Filter.map_comap_setCoe_val 📋 Mathlib.Order.Filter.Map
  {β : Type u_2} (f : Filter β) (s : Set β) : Filter.map Subtype.val (Filter.comap Subtype.val f) = f ⊓ Filter.principal s
• Filter.subtype_coe_map_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} (s : Set α) (f : Filter α) : Filter.map Subtype.val (Filter.comap Subtype.val f) = f ⊓ Filter.principal s
• Filter.disjoint_comap_iff 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {g₁ g₂ : Filter β} {m : α → β} (h : Function.Surjective m) : Disjoint (Filter.comap m g₁) (Filter.comap m g₂) ↔ Disjoint g₁ g₂
• Filter.mem_comap_prodMk 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {x : α} {s : Set β} {F : Filter (α × β)} : s ∈ Filter.comap (Prod.mk x) F ↔ {p | p.1 = x → p.2 ∈ s} ∈ F
• Filter.inf_principal_eq_bot_iff_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {F : Filter α} {s : Set α} : F ⊓ Filter.principal s = ⊥ ↔ Filter.comap Subtype.val F = ⊥
• Filter.image_coe_mem_of_mem_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {f : Filter α} {U : Set α} (h : U ∈ f) {W : Set ↑U} (W_in : W ∈ Filter.comap Subtype.val f) : Subtype.val '' W ∈ f
• Filter.map_swap4_eq_comap 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f : Filter ((α × β) × γ × δ)} : Filter.map (fun p => ((p.1.1, p.2.1), p.1.2, p.2.2)) f = Filter.comap (fun p => ((p.1.1, p.2.1), p.1.2, p.2.2)) f
• Filter.comap_hasBasis 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {β : Type u_2} (f : α → β) (l : Filter β) : (Filter.comap f l).HasBasis (fun s => s ∈ l) fun s => f ⁻¹' s
• Filter.HasAntitoneBasis.comap 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {β : Type u_2} {ι'' : Type u_5} [Preorder ι''] {l : Filter α} {s : ι'' → Set α} (hf : l.HasAntitoneBasis s) (m : β → α) : (Filter.comap m l).HasAntitoneBasis fun x => m ⁻¹' s x
• Filter.HasBasis.comap 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {β : Type u_2} {ι : Sort u_3} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (f : β → α) (hl : l.HasBasis p s) : (Filter.comap f l).HasBasis p fun i => f ⁻¹' s i
• Filter.map_sigma_mk_comap 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {β : Type u_2} {π : α → Type u_5} {π' : β → Type u_6} {f : α → β} (hf : Function.Injective f) (g : (a : α) → π a → π' (f a)) (a : α) (l : Filter (π' (f a))) : Filter.map (Sigma.mk a) (Filter.comap (g a) l) = Filter.comap (Sigma.map f g) (Filter.map (Sigma.mk (f a)) l)
• Filter.tendsto_comap 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {f : α → β} {x : Filter β} : Filter.Tendsto f (Filter.comap f x) x
• Filter.Tendsto.le_comap 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {f : α → β} {l₁ : Filter α} {l₂ : Filter β} : Filter.Tendsto f l₁ l₂ → l₁ ≤ Filter.comap f l₂
• Filter.tendsto_iff_comap 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {f : α → β} {l₁ : Filter α} {l₂ : Filter β} : Filter.Tendsto f l₁ l₂ ↔ l₁ ≤ Filter.comap f l₂
• Filter.tendsto_comap_iff 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {g : β → γ} {a : Filter α} {c : Filter γ} : Filter.Tendsto f a (Filter.comap g c) ↔ Filter.Tendsto (g ∘ f) a c
• Filter.comap_eq_of_inverse 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {f : Filter α} {g : Filter β} {φ : α → β} (ψ : β → α) (eq : ψ ∘ φ = id) (hφ : Filter.Tendsto φ f g) (hψ : Filter.Tendsto ψ g f) : Filter.comap φ g = f
• Filter.tendsto_comap'_iff 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : α → β} {f : Filter α} {g : Filter β} {i : γ → α} (h : Set.range i ∈ f) : Filter.Tendsto (m ∘ i) (Filter.comap i f) g ↔ Filter.Tendsto m f g
• Filter.Tendsto.of_tendsto_comp 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {g : β → γ} {a : Filter α} {b : Filter β} {c : Filter γ} (hfg : Filter.Tendsto (g ∘ f) a c) (hg : Filter.comap g c ≤ b) : Filter.Tendsto f a b
• Filter.comap_embedding_atBot 📋 Mathlib.Order.Filter.AtTopBot.Tendsto
  {β : Type u_4} {γ : Type u_5} [Preorder β] [Preorder γ] {e : β → γ} (hm : ∀ (b₁ b₂ : β), e b₂ ≤ e b₁ ↔ b₂ ≤ b₁) (hu : ∀ (c : γ), ∃ b, e b ≤ c) : Filter.comap e Filter.atBot = Filter.atBot
• Filter.comap_embedding_atTop 📋 Mathlib.Order.Filter.AtTopBot.Tendsto
  {β : Type u_4} {γ : Type u_5} [Preorder β] [Preorder γ] {e : β → γ} (hm : ∀ (b₁ b₂ : β), e b₁ ≤ e b₂ ↔ b₁ ≤ b₂) (hu : ∀ (c : γ), ∃ b, c ≤ e b) : Filter.comap e Filter.atTop = Filter.atTop
• Filter.atBot_Iic_eq 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} [Preorder α] [IsCodirectedOrder α] (a : α) : Filter.atBot = Filter.comap Subtype.val Filter.atBot
• Filter.atBot_Iio_eq 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} [Preorder α] [IsCodirectedOrder α] (a : α) : Filter.atBot = Filter.comap Subtype.val Filter.atBot
• Filter.atTop_Ici_eq 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} [Preorder α] [IsDirectedOrder α] (a : α) : Filter.atTop = Filter.comap Subtype.val Filter.atTop
• Filter.atTop_Ioi_eq 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} [Preorder α] [IsDirectedOrder α] (a : α) : Filter.atTop = Filter.comap Subtype.val Filter.atTop
• Filter.comap.isCountablyGenerated 📋 Mathlib.Order.Filter.CountablyGenerated
  {α : Type u_1} {β : Type u_2} (l : Filter β) [l.IsCountablyGenerated] (f : α → β) : (Filter.comap f l).IsCountablyGenerated
• Filter.ker_comap 📋 Mathlib.Order.Filter.Ker
  {α : Type u_2} {β : Type u_3} (m : α → β) (f : Filter β) : (Filter.comap m f).ker = m ⁻¹' f.ker
• Filter.pi_comap 📋 Mathlib.Order.Filter.Pi
  {ι : Type u_1} {α : ι → Type u_2} {β : ι → Type u_3} {f : (i : ι) → α i → β i} {l : (i : ι) → Filter (β i)} : (Filter.pi fun i => Filter.comap (f i) (l i)) = Filter.comap (Pi.map f) (Filter.pi l)
• Filter.bot_coprod 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} (l : Filter β) : ⊥.coprod l = Filter.comap Prod.snd l
• Filter.coprod_bot 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} (l : Filter α) : l.coprod ⊥ = Filter.comap Prod.fst l
• Filter.prod_top 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {f : Filter α} : f ×ˢ ⊤ = Filter.comap Prod.fst f
• Filter.top_prod 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {g : Filter β} : ⊤ ×ˢ g = Filter.comap Prod.snd g
• Filter.prod_comm' 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {f : Filter α} {g : Filter β} : f ×ˢ g = Filter.comap Prod.swap (g ×ˢ f)
• Filter.comap_prod 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β × γ) (b : Filter β) (c : Filter γ) : Filter.comap f (b ×ˢ c) = Filter.comap (Prod.fst ∘ f) b ⊓ Filter.comap (Prod.snd ∘ f) c
• Filter.comap_prodMap_prod 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : α → β) (g : γ → δ) (lb : Filter β) (ld : Filter δ) : Filter.comap (Prod.map f g) (lb ×ˢ ld) = Filter.comap f lb ×ˢ Filter.comap g ld
• Filter.prod_comap_comap_eq 📋 Mathlib.Order.Filter.Prod
  {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x} {f₁ : Filter α₁} {f₂ : Filter α₂} {m₁ : β₁ → α₁} {m₂ : β₂ → α₂} : Filter.comap m₁ f₁ ×ˢ Filter.comap m₂ f₂ = Filter.comap (fun p => (m₁ p.1, m₂ p.2)) (f₁ ×ˢ f₂)
• Function.Injective.comap_cofinite_eq 📋 Mathlib.Order.Filter.Cofinite
  {α : Type u_2} {β : Type u_3} {f : α → β} (hf : Function.Injective f) : Filter.comap f Filter.cofinite = Filter.cofinite
• Filter.comap_cofinite_le 📋 Mathlib.Order.Filter.Cofinite
  {α : Type u_2} {β : Type u_3} (f : α → β) : Filter.comap f Filter.cofinite ≤ Filter.cofinite
• OrderIso.comap_atBot 📋 Mathlib.Order.Filter.AtTopBot.Map
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) : Filter.comap (⇑e) Filter.atBot = Filter.atBot
• OrderIso.comap_atTop 📋 Mathlib.Order.Filter.AtTopBot.Map
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ≃o β) : Filter.comap (⇑e) Filter.atTop = Filter.atTop
• Ultrafilter.comap_inf_principal_neBot_of_image_mem 📋 Mathlib.Order.Filter.Ultrafilter.Defs
  {α : Type u} {β : Type v} {m : α → β} {s : Set α} {g : Ultrafilter β} (h : m '' s ∈ g) : (Filter.comap m ↑g ⊓ Filter.principal s).NeBot
• Ultrafilter.coe_comap 📋 Mathlib.Order.Filter.Ultrafilter.Defs
  {α : Type u} {β : Type v} {m : α → β} (u : Ultrafilter β) (inj : Function.Injective m) (large : Set.range m ∈ u) : ↑(u.comap inj large) = Filter.comap m ↑u
• Filter.comap_inv 📋 Mathlib.Order.Filter.Pointwise
  {α : Type u_2} [InvolutiveInv α] {f : Filter α} : Filter.comap Inv.inv f = f⁻¹
• Filter.comap_neg 📋 Mathlib.Order.Filter.Pointwise
  {α : Type u_2} [InvolutiveNeg α] {f : Filter α} : Filter.comap Neg.neg f = -f
• Filter.comap_add_comap_le 📋 Mathlib.Order.Filter.Pointwise
  {F : Type u_1} {α : Type u_2} {β : Type u_3} [AddZeroClass α] [AddZeroClass β] [FunLike F α β] [AddHomClass F α β] (m : F) {f g : Filter β} : Filter.comap (⇑m) f + Filter.comap (⇑m) g ≤ Filter.comap (⇑m) (f + g)
• Filter.comap_mul_comap_le 📋 Mathlib.Order.Filter.Pointwise
  {F : Type u_1} {α : Type u_2} {β : Type u_3} [MulOneClass α] [MulOneClass β] [FunLike F α β] [MulHomClass F α β] (m : F) {f g : Filter β} : Filter.comap (⇑m) f * Filter.comap (⇑m) g ≤ Filter.comap (⇑m) (f * g)
• Filter.comap_eq_lift' 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} {f : Filter β} {m : α → β} : Filter.comap m f = f.lift' (Set.preimage m)
• Filter.comap_lift'_eq 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : Filter α} {h : Set α → Set β} {m : γ → β} : Filter.comap m (f.lift' h) = f.lift' (Set.preimage m ∘ h)
• Filter.comap_lift_eq 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : Filter α} {g : Set α → Filter β} {m : γ → β} : Filter.comap m (f.lift g) = f.lift (Filter.comap m ∘ g)
• Filter.comap_lift_eq2 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : Filter α} {m : β → α} {g : Set β → Filter γ} (hg : Monotone g) : (Filter.comap m f).lift g = f.lift (g ∘ Set.preimage m)
• Filter.comap_lift'_eq2 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : Filter α} {m : β → α} {g : Set β → Set γ} (hg : Monotone g) : (Filter.comap m f).lift' g = f.lift' (g ∘ Set.preimage m)
• mem_closure_iff_comap_neBot 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {x : X} {s : Set X} : x ∈ closure s ↔ (Filter.comap Subtype.val (nhds x)).NeBot
• nhds_induced 📋 Mathlib.Topology.Order
  {α : Type u} {β : Type v} [T : TopologicalSpace α] (f : β → α) (a : β) : nhds a = Filter.comap f (nhds (f a))
• induced_iff_nhds_eq 📋 Mathlib.Topology.Order
  {α : Type u} {β : Type v} [tα : TopologicalSpace α] [tβ : TopologicalSpace β] (f : β → α) : tβ = TopologicalSpace.induced f tα ↔ ∀ (b : β), nhds b = Filter.comap f (nhds (f b))
• Topology.IsInducing.nhds_eq_comap 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) (x : X) : nhds x = Filter.comap f (nhds (f x))
• IsOpenMap.accPt_comap 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) {x : X} {l : Filter Y} (h : AccPt (f x) l) : AccPt x (Filter.comap f l)
• IsOpenMap.clusterPt_comap 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) {x : X} {l : Filter Y} (h : ClusterPt (f x) l) : ClusterPt x (Filter.comap f l)
• Topology.isInducing_iff_nhds 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] : Topology.IsInducing f ↔ ∀ (x : X), nhds x = Filter.comap f (nhds (f x))
• isOpenMap_iff_clusterPt_comap 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsOpenMap f ↔ ∀ (x : X) (l : Filter Y), ClusterPt (f x) l → ClusterPt x (Filter.comap f l)
• Topology.IsOpenEmbedding.accPt_comap_iff 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Topology.IsOpenEmbedding f) {x : X} {l : Filter Y} : AccPt x (Filter.comap f l) ↔ AccPt (f x) l
• Topology.IsEmbedding.mk' 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (inj : Function.Injective f) (induced : ∀ (x : X), Filter.comap f (nhds (f x)) = nhds x) : Topology.IsEmbedding f
• Topology.IsInducing.nhdsSet_eq_comap 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : Topology.IsInducing f) (s : Set X) : nhdsSet s = Filter.comap f (nhdsSet (f '' s))
• IsOpenMap.clusterPt_comap_iff 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) (hfc : Continuous f) {x : X} {l : Filter Y} : ClusterPt x (Filter.comap f l) ↔ ClusterPt (f x) l
• IsClosedMap.comap_nhdsSet_eq 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedMap f) (hf' : Continuous f) (s : Set Y) : Filter.comap f (nhdsSet s) = nhdsSet (f ⁻¹' s)
• IsClosedMap.comap_nhdsSet_le 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsClosedMap f → ∀ {s : Set Y}, Filter.comap f (nhdsSet s) ≤ nhdsSet (f ⁻¹' s)
• isClosedMap_iff_comap_nhdsSet_le 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsClosedMap f ↔ ∀ {s : Set Y}, Filter.comap f (nhdsSet s) ≤ nhdsSet (f ⁻¹' s)
• IsClosedMap.comap_nhds_eq 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedMap f) (hf' : Continuous f) (y : Y) : Filter.comap f (nhds y) = nhdsSet (f ⁻¹' {y})
• IsClosedMap.comap_nhds_le 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsClosedMap f → ∀ {y : Y}, Filter.comap f (nhds y) ≤ nhdsSet (f ⁻¹' {y})
• isClosedMap_iff_comap_nhds_le 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsClosedMap f ↔ ∀ {y : Y}, Filter.comap f (nhds y) ≤ nhdsSet (f ⁻¹' {y})
• Homeomorph.nhds_eq_comap 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (x : X) : nhds x = Filter.comap (⇑h) (nhds (h x))
• Homeomorph.comap_nhds_eq 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (y : Y) : Filter.comap (⇑h) (nhds y) = nhds (h.symm y)
• comap_nhdsWithin_range 📋 Mathlib.Topology.Constructions
  {α : Type u_2} {β : Type u_3} [TopologicalSpace β] (f : α → β) (y : β) : Filter.comap f (nhdsWithin y (Set.range f)) = Filter.comap f (nhds y)
• nhds_subtype_eq_comap 📋 Mathlib.Topology.Constructions
  {X : Type u} [TopologicalSpace X] {p : X → Prop} {x : X} {h : p x} : nhds ⟨x, h⟩ = Filter.comap Subtype.val (nhds x)
• comap_sigmaMk_nhds 📋 Mathlib.Topology.Constructions
  {ι : Type u_2} {σ : ι → Type u_4} [(i : ι) → TopologicalSpace (σ i)] (i : ι) (x : σ i) : Filter.comap (Sigma.mk i) (nhds ⟨i, x⟩) = nhds x
• nhds_subtype 📋 Mathlib.Topology.Constructions
  {X : Type u} [TopologicalSpace X] (s : Set X) (x : { x // x ∈ s }) : nhds x = Filter.comap Subtype.val (nhds ↑x)
• nhds_subtype_eq_comap_nhdsWithin 📋 Mathlib.Topology.Constructions
  {X : Type u} [TopologicalSpace X] (s : Set X) (x : { x // x ∈ s }) : nhds x = Filter.comap Subtype.val (nhdsWithin (↑x) s)
• nhdsSet_induced 📋 Mathlib.Topology.NhdsWithin
  {α : Type u_3} {β : Type u_4} {t : TopologicalSpace β} (f : α → β) (s : Set α) : nhdsSet s = Filter.comap f (nhdsSet (f '' s))
• nhdsWithin_pi_univ_eq 📋 Mathlib.Topology.NhdsWithin
  {ι : Type u_3} {X : ι → Type u_4} [(i : ι) → TopologicalSpace (X i)] [Finite ι] (s : (i : ι) → Set (X i)) (x : (i : ι) → X i) : nhdsWithin x (Set.univ.pi s) = ⨅ i, Filter.comap (fun x => x i) (nhdsWithin (x i) (s i))
• nhdsWithin_pi_eq' 📋 Mathlib.Topology.NhdsWithin
  {ι : Type u_3} {X : ι → Type u_4} [(i : ι) → TopologicalSpace (X i)] {I : Set ι} (hI : I.Finite) (s : (i : ι) → Set (X i)) (x : (i : ι) → X i) : nhdsWithin x (I.pi s) = ⨅ i, Filter.comap (fun x => x i) (nhds (x i) ⊓ ⨅ (_ : i ∈ I), Filter.principal (s i))
• nhdsWithin_subtype 📋 Mathlib.Topology.NhdsWithin
  {α : Type u_1} [TopologicalSpace α] (s : Set α) (a : { x // x ∈ s }) (t : Set { x // x ∈ s }) : nhdsWithin a t = Filter.comap Subtype.val (nhdsWithin (↑a) (Subtype.val '' t))
• nhdsWithin_pi_eq 📋 Mathlib.Topology.NhdsWithin
  {ι : Type u_3} {X : ι → Type u_4} [(i : ι) → TopologicalSpace (X i)] {I : Set ι} (hI : I.Finite) (s : (i : ι) → Set (X i)) (x : (i : ι) → X i) : nhdsWithin x (I.pi s) = (⨅ i ∈ I, Filter.comap (fun x => x i) (nhdsWithin (x i) (s i))) ⊓ ⨅ i, ⨅ (_ : i ∉ I), Filter.comap (fun x => x i) (nhds (x i))
• mem_nhdsWithin_subtype 📋 Mathlib.Topology.NhdsWithin
  {α : Type u_1} [TopologicalSpace α] {s : Set α} {a : { x // x ∈ s }} {t u : Set { x // x ∈ s }} : t ∈ nhdsWithin a u ↔ t ∈ Filter.comap Subtype.val (nhdsWithin (↑a) (Subtype.val '' u))
• nhdsWithin_le_comap 📋 Mathlib.Topology.ContinuousOn
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} {s : Set α} {x : α} (ctsf : ContinuousWithinAt f s x) : nhdsWithin x s ≤ Filter.comap f (nhdsWithin (f x) (f '' s))
• Bornology.comap_cobounded_le_iff 📋 Mathlib.Topology.Bornology.Basic
  {α : Type u_2} {β : Type u_3} {x✝ : Bornology α} [Bornology β] {f : α → β} : Filter.comap f (Bornology.cobounded β) ≤ Bornology.cobounded α ↔ ∀ ⦃s : Set α⦄, Bornology.IsBounded s → Bornology.IsBounded (f '' s)
• Filter.comap_cocompact_le 📋 Mathlib.Topology.Compactness.Compact
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) : Filter.comap f (Filter.cocompact Y) ≤ Filter.cocompact X
• Filter.smallSets_comap_eq_comap_image 📋 Mathlib.Order.Filter.SmallSets
  {α : Type u_1} {β : Type u_2} (l : Filter β) (f : α → β) : (Filter.comap f l).smallSets = Filter.comap (Set.image f) l.smallSets
• Filter.smallSets_comap 📋 Mathlib.Order.Filter.SmallSets
  {α : Type u_1} {β : Type u_2} (l : Filter β) (f : α → β) : (Filter.comap f l).smallSets = l.lift' (Set.powerset ∘ Set.preimage f)
• Filter.comap_smallSets 📋 Mathlib.Order.Filter.SmallSets
  {α : Type u_1} {β : Type u_2} (l : Filter β) (f : α → Set β) : Filter.comap f l.smallSets = l.lift' (Set.preimage f ∘ Set.powerset)
• SeparationQuotient.comap_mk_nhds_mk 📋 Mathlib.Topology.Inseparable
  {X : Type u_1} [TopologicalSpace X] {x : X} : Filter.comap SeparationQuotient.mk (nhds (SeparationQuotient.mk x)) = nhds x
• SeparationQuotient.comap_mk_nhdsSet_image 📋 Mathlib.Topology.Inseparable
  {X : Type u_1} [TopologicalSpace X] {s : Set X} : Filter.comap SeparationQuotient.mk (nhdsSet (SeparationQuotient.mk '' s)) = nhdsSet s
• SeparationQuotient.comap_mk_nhdsSet 📋 Mathlib.Topology.Inseparable
  {X : Type u_1} [TopologicalSpace X] {t : Set (SeparationQuotient X)} : Filter.comap SeparationQuotient.mk (nhdsSet t) = nhdsSet (SeparationQuotient.mk ⁻¹' t)
• instCountableInterFilterComap 📋 Mathlib.Order.Filter.CountableInter
  {α : Type u_2} {β : Type u_3} (l : Filter β) [CountableInterFilter l] (f : α → β) : CountableInterFilter (Filter.comap f l)
• Filter.comap_coLindelof_le 📋 Mathlib.Topology.Compactness.Lindelof
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) : Filter.comap f (Filter.coLindelof Y) ≤ Filter.coLindelof X
• IsDenseInducing.comap_nhds_neBot 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (di : IsDenseInducing i) (b : β) : (Filter.comap i (nhds b)).NeBot
• IsDenseInducing.nhds_eq_comap 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} (di : IsDenseInducing i) (a : α) : nhds a = Filter.comap i (nhds (i a))
• Dense.comap_val_nhds_neBot 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} [TopologicalSpace α] {s : Set α} (hs : Dense s) (a : α) : (Filter.comap Subtype.val (nhds a)).NeBot
• IsDenseInducing.extend_eq_of_tendsto 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace γ] [T2Space γ] (di : IsDenseInducing i) {b : β} {c : γ} {f : α → γ} (hf : Filter.Tendsto f (Filter.comap i (nhds b)) (nhds c)) : di.extend f b = c
• IsDenseInducing.continuous_extend 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace γ] [T3Space γ] {f : α → γ} (di : IsDenseInducing i) (hf : ∀ (b : β), ∃ c, Filter.Tendsto f (Filter.comap i (nhds b)) (nhds c)) : Continuous (di.extend f)
• IsDenseInducing.extend_eq' 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace γ] [T2Space γ] {f : α → γ} (di : IsDenseInducing i) (hf : ∀ (b : β), ∃ c, Filter.Tendsto f (Filter.comap i (nhds b)) (nhds c)) (a : α) : di.extend f (i a) = f a
• Dense.extend_eq_of_tendsto 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} {f : ↑s → β} [T2Space β] (hs : Dense s) {a : α} {b : β} (hf : Filter.Tendsto f (Filter.comap Subtype.val (nhds a)) (nhds b)) : hs.extend f a = b
• IsDenseInducing.continuousAt_extend 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace γ] [T3Space γ] {b : β} {f : α → γ} (di : IsDenseInducing i) (hf : ∀ᶠ (x : β) in nhds b, ∃ c, Filter.Tendsto f (Filter.comap i (nhds x)) (nhds c)) : ContinuousAt (di.extend f) b
• IsDenseInducing.extend_unique_at 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace γ] [T2Space γ] {b : β} {f : α → γ} {g : β → γ} (di : IsDenseInducing i) (hf : ∀ᶠ (x : α) in Filter.comap i (nhds b), g (i x) = f x) (hg : ContinuousAt g b) : di.extend f b = g b
• Dense.continuous_extend 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} {f : ↑s → β} [T3Space β] (hs : Dense s) (hf : ∀ (a : α), ∃ b, Filter.Tendsto f (Filter.comap Subtype.val (nhds a)) (nhds b)) : Continuous (hs.extend f)
• IsDenseInducing.tendsto_comap_nhds_nhds 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] {i : α → β} [TopologicalSpace δ] {f : γ → α} {g : γ → δ} {h : δ → β} {d : δ} {a : α} (di : IsDenseInducing i) (H : Filter.Tendsto h (nhds d) (nhds (i a))) (comm : h ∘ g = i ∘ f) : Filter.Tendsto f (Filter.comap g (nhds d)) (nhds a)
• Dense.continuousAt_extend 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} {f : ↑s → β} [T3Space β] {a : α} (hs : Dense s) (hf : ∀ᶠ (x : α) in nhds a, ∃ b, Filter.Tendsto f (Filter.comap Subtype.val (nhds x)) (nhds b)) : ContinuousAt (hs.extend f) a
• Dense.extend_unique_at 📋 Mathlib.Topology.DenseEmbedding
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {s : Set α} {f : ↑s → β} [T2Space β] {a : α} {g : α → β} (hs : Dense s) (hf : ∀ᶠ (x : ↑s) in Filter.comap Subtype.val (nhds a), g ↑x = f x) (hg : ContinuousAt g a) : hs.extend f a = g a
• Homeomorph.comap_coclosedCompact 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) : Filter.comap (⇑h) (Filter.coclosedCompact Y) = Filter.coclosedCompact X
• Homeomorph.comap_cocompact 📋 Mathlib.Topology.Homeomorph.Lemmas
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) : Filter.comap (⇑h) (Filter.cocompact Y) = Filter.cocompact X
• AddOpposite.comap_op_nhds 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_1} [TopologicalSpace M] (x : Mᵃᵒᵖ) : Filter.comap AddOpposite.op (nhds x) = nhds (AddOpposite.unop x)
• AddOpposite.comap_unop_nhds 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_1} [TopologicalSpace M] (x : M) : Filter.comap AddOpposite.unop (nhds x) = nhds (AddOpposite.op x)
• MulOpposite.comap_op_nhds 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_1} [TopologicalSpace M] (x : Mᵐᵒᵖ) : Filter.comap MulOpposite.op (nhds x) = nhds (MulOpposite.unop x)
• MulOpposite.comap_unop_nhds 📋 Mathlib.Topology.Algebra.Constructions
  {M : Type u_1} [TopologicalSpace M] (x : M) : Filter.comap MulOpposite.unop (nhds x) = nhds (MulOpposite.op x)
• comap_swap_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] : Filter.comap Prod.swap (uniformity α) = uniformity α
• nhds_eq_comap_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] {x : α} : nhds x = Filter.comap (Prod.mk x) (uniformity α)
• UniformSpace.nhds_eq_comap_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type u} [self : UniformSpace α] (x : α) : nhds x = Filter.comap (Prod.mk x) UniformSpace.uniformity

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c