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Result

Found 539 declarations mentioning Filter.map. Of these, only the first 200 are shown.

• Filter.map 📋 Mathlib.Order.Filter.Defs
  {α : Type u_1} {β : Type u_2} (m : α → β) (f : Filter α) : Filter β
• Filter.map_id 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {f : Filter α} : Filter.map id f = f
• Filter.map_id' 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {f : Filter α} : Filter.map (fun x => x) f = f
• Filter.map_neBot 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} [hf : f.NeBot] : (Filter.map m f).NeBot
• Filter.NeBot.map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} (hf : f.NeBot) (m : α → β) : (Filter.map m f).NeBot
• Filter.NeBot.of_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} : (Filter.map m f).NeBot → f.NeBot
• Filter.map_injective 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {m : α → β} (hm : Function.Injective m) : Function.Injective (Filter.map m)
• Filter.map_neBot_iff 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (f : α → β) {F : Filter α} : (Filter.map f F).NeBot ↔ F.NeBot
• Function.Commute.filter_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {f g : α → α} (h : Function.Commute f g) : Function.Commute (Filter.map f) (Filter.map g)
• Filter.map_bot 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {m : α → β} : Filter.map m ⊥ = ⊥
• Filter.map_def 📋 Mathlib.Order.Filter.Map
  {α β : Type u_6} (m : α → β) (f : Filter α) : m <$> f = Filter.map m f
• Filter.map_const 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} [f.NeBot] {c : β} : Filter.map (fun x => c) f = pure c
• Filter.map_top 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (f : α → β) : Filter.map f ⊤ = Filter.principal (Set.range f)
• Filter.range_mem_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} : Set.range m ∈ Filter.map m f
• 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.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.map_mono 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {m : α → β} : Monotone (Filter.map m)
• Filter.map_principal 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {s : Set α} {f : α → β} : Filter.map f (Filter.principal s) = Filter.principal (f '' s)
• Filter.map_pure 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (f : α → β) (a : α) : Filter.map f (pure a) = pure (f a)
• Filter.pure_seq_eq_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (g : α → β) (f : Filter α) : (pure g).seq f = Filter.map g f
• Function.LeftInverse.filter_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} {g : β → α} (hfg : Function.LeftInverse g f) : Function.LeftInverse (Filter.map g) (Filter.map f)
• Function.RightInverse.filter_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} {g : β → α} (hfg : Function.RightInverse g f) : Function.RightInverse (Filter.map g) (Filter.map f)
• Function.Surjective.filter_map_top 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : α → β} (hf : Function.Surjective f) : Filter.map f ⊤ = ⊤
• Filter.eventually_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} {P : β → Prop} : (∀ᶠ (b : β) in Filter.map m f, P b) ↔ ∀ᶠ (a : α) in f, P (m a)
• Filter.frequently_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} {P : β → Prop} : (∃ᶠ (b : β) in Filter.map m f, P b) ↔ ∃ᶠ (a : α) in f, P (m a)
• 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.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.map_congr 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {m₁ m₂ : α → β} {f : Filter α} (h : m₁ =ᶠ[f] m₂) : Filter.map m₁ f = Filter.map m₂ f
• Filter.seq_pure 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (f : Filter (α → β)) (a : α) : f.seq (pure a) = Filter.map (fun g => g a) f
• Filter.canLift 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (c : β → α) (p : α → Prop) [CanLift α β c p] : CanLift (Filter α) (Filter β) (Filter.map c) fun f => ∀ᶠ (x : α) in f, p x
• Filter.bot_eq_map_iff 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} : ⊥ = Filter.map m f ↔ f = ⊥
• 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.map_eq_bot_iff 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} : Filter.map m f = ⊥ ↔ f = ⊥
• 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.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.map_inj 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f g : Filter α} {m : α → β} (hm : Function.Injective m) : Filter.map m f = Filter.map m g ↔ f = g
• Filter.map_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : Filter α} {m : α → β} {m' : β → γ} : Filter.map m' (Filter.map m f) = Filter.map (m' ∘ m) f
• Function.Semiconj.filter_map 📋 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.map f) (Filter.map ga) (Filter.map gb)
• Set.InjOn.filter_map_Iic 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {s : Set α} {m : α → β} : Set.InjOn m s → Set.InjOn (Filter.map m) (Set.Iic (Filter.principal s))
• Filter.filter_injOn_Iic_iff_injOn 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {s : Set α} {m : α → β} : Set.InjOn (Filter.map m) (Set.Iic (Filter.principal s)) ↔ Set.InjOn m s
• Filter.bind_map 📋 Mathlib.Order.Filter.Map
  {γ : Type u_3} {α : Type u_6} {β : Type u_7} (m : α → β) (f : Filter α) (g : β → Filter γ) : (Filter.map m f).bind g = f.bind (g ∘ m)
• 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.image_mem_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} {s : Set α} (hs : s ∈ f) : m '' s ∈ Filter.map m f
• Filter.mem_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} {t : Set β} : t ∈ Filter.map m f ↔ m ⁻¹' t ∈ f
• 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.map_compose 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : α → β} {m' : β → γ} : Filter.map m' ∘ Filter.map m = Filter.map (m' ∘ m)
• Filter.map_bind 📋 Mathlib.Order.Filter.Map
  {γ : Type u_3} {α : Type u_6} {β : Type u_7} (m : β → γ) (f : Filter α) (g : α → Filter β) : Filter.map m (f.bind g) = f.bind (Filter.map m ∘ g)
• Filter.map_iSup 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {ι : Sort u_5} {m : α → β} {f : ι → Filter α} : Filter.map m (⨆ i, f i) = ⨆ i, Filter.map m (f i)
• Filter.map_sup 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f₁ f₂ : Filter α} {m : α → β} : Filter.map m (f₁ ⊔ f₂) = Filter.map m f₁ ⊔ Filter.map m f₂
• 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_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : Filter α} {m : α → β} {f₁ f₂ : β → γ} : f₁ =ᶠ[Filter.map m f] f₂ ↔ f₁ ∘ m =ᶠ[f] f₂ ∘ m
• Set.LeftInvOn.filter_map_Iic 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {s : Set α} {f : α → β} {g : β → α} (hfg : Set.LeftInvOn g f s) : Set.LeftInvOn (Filter.map g) (Filter.map f) (Set.Iic (Filter.principal s))
• Set.RightInvOn.filter_map_Iic 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {t : Set β} {f : α → β} {g : β → α} (hfg : Set.RightInvOn g f t) : Set.RightInvOn (Filter.map g) (Filter.map f) (Set.Iic (Filter.principal t))
• Filter.image_mem_map_iff 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} {s : Set α} (hf : Function.Injective m) : m '' s ∈ Filter.map m f ↔ s ∈ f
• Filter.map_generate_le_generate_preimage_preimage 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (U : Set (Set β)) (f : β → α) : Filter.map f (Filter.generate U) ≤ Filter.generate ((fun x => f ⁻¹' x) ⁻¹' U)
• Filter.map_inf 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f g : Filter α} {m : α → β} (h : Function.Injective m) : Filter.map m (f ⊓ g) = Filter.map m f ⊓ Filter.map m g
• Filter.eventuallyLE_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : Filter α} {m : α → β} [LE γ] {f₁ f₂ : β → γ} : f₁ ≤ᶠ[Filter.map m f] f₂ ↔ f₁ ∘ m ≤ᶠ[f] f₂ ∘ m
• Filter.mem_map' 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} {t : Set β} : t ∈ Filter.map m f ↔ {x | m x ∈ t} ∈ 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_iInf_le 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {ι : Sort u_5} {f : ι → Filter α} {m : α → β} : Filter.map m (iInf f) ≤ ⨅ i, Filter.map m (f i)
• 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.map_inf_le 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f g : Filter α} {m : α → β} : Filter.map m (f ⊓ g) ≤ Filter.map m f ⊓ Filter.map m g
• Set.SurjOn.filter_map_Iic 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {m : α → β} : Set.SurjOn m s t → Set.SurjOn (Filter.map m) (Set.Iic (Filter.principal s)) (Set.Iic (Filter.principal t))
• Filter.map_surjOn_Iic_iff_surjOn 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {m : α → β} : Set.SurjOn (Filter.map m) (Set.Iic (Filter.principal s)) (Set.Iic (Filter.principal t)) ↔ Set.SurjOn m s t
• Filter.map_inl_inf_map_inr 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {g : Filter β} : Filter.map Sum.inl f ⊓ Filter.map Sum.inr g = ⊥
• Filter.map_inr_inf_map_inl 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {g : Filter β} : Filter.map Sum.inr f ⊓ Filter.map Sum.inl g = ⊥
• Filter.map_le_map_iff 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f g : Filter α} {m : α → β} (hm : Function.Injective m) : Filter.map m f ≤ Filter.map m g ↔ f ≤ g
• Filter.prod_map_seq_comm 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} (f : Filter α) (g : Filter β) : (Filter.map Prod.mk f).seq g = (Filter.map (fun b a => (a, b)) g).seq f
• Filter.mem_map_iff_exists_image 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} {t : Set β} : t ∈ Filter.map m f ↔ ∃ s ∈ f, m '' s ⊆ t
• Filter.le_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} {g : Filter β} (h : ∀ s ∈ f, m '' s ∈ g) : g ≤ Filter.map m f
• Filter.le_map_iff 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f : Filter α} {m : α → β} {g : Filter β} : g ≤ Filter.map m f ↔ ∀ s ∈ f, m '' s ∈ g
• Filter.map_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) (F : Filter α) : Filter.map ψ (Filter.map φ F) = Filter.map ρ (Filter.map θ F)
• Filter.seq_assoc 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} (x : Filter α) (g : Filter (α → β)) (h : Filter (β → γ)) : h.seq (g.seq x) = ((Filter.map (fun x1 x2 => x1 ∘ x2) h).seq g).seq x
• Filter.map_eq_map_iff_of_injOn 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f g : Filter α} {m : α → β} {s : Set α} (hsf : s ∈ f) (hsg : s ∈ g) (hm : Set.InjOn m s) : Filter.map m f = Filter.map m g ↔ f = g
• Filter.map_surjOn_Iic_iff_le_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {F : Filter α} {G : Filter β} {m : α → β} : Set.SurjOn (Filter.map m) (Set.Iic F) (Set.Iic G) ↔ G ≤ Filter.map m 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.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.principal_eq_map_coe_top 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} (s : Set α) : Filter.principal s = Filter.map Subtype.val ⊤
• Filter.map_iInf_eq 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {ι : Sort u_5} {f : ι → Filter α} {m : α → β} (hf : Directed (fun x1 x2 => x1 ≥ x2) f) [Nonempty ι] : Filter.map m (iInf f) = ⨅ i, Filter.map m (f i)
• 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_of_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_6} {β : Type u_7} {F G : Filter α} {f : α → β} (h : Disjoint (Filter.map f F) (Filter.map f G)) : Disjoint F G
• Filter.map_inf' 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {f g : Filter α} {m : α → β} {t : Set α} (htf : t ∈ f) (htg : t ∈ g) (h : Set.InjOn m t) : Filter.map m (f ⊓ g) = Filter.map m f ⊓ Filter.map m 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 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_map 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {m : α → β} (hm : Function.Injective m) {f₁ f₂ : Filter α} : Disjoint (Filter.map m f₁) (Filter.map m f₂) ↔ Disjoint f₁ f₂
• Filter.map_le_map_iff_of_injOn 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {l₁ l₂ : Filter α} {f : α → β} {s : Set α} (h₁ : s ∈ l₁) (h₂ : s ∈ l₂) (hinj : Set.InjOn f s) : Filter.map f l₁ ≤ Filter.map f l₂ ↔ l₁ ≤ l₂
• Filter.mem_map_seq_iff 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : Filter α} {g : Filter β} {m : α → β → γ} {s : Set γ} : s ∈ (Filter.map m f).seq g ↔ ∃ t u, t ∈ g ∧ u ∈ f ∧ ∀ x ∈ u, ∀ y ∈ t, m x y ∈ s
• Filter.map_biInf_eq 📋 Mathlib.Order.Filter.Map
  {α : Type u_1} {β : Type u_2} {ι : Type w} {f : ι → Filter α} {m : α → β} {p : ι → Prop} (h : DirectedOn (f ⁻¹'o fun x1 x2 => x1 ≥ x2) {x | p x}) (ne : ∃ i, p i) : Filter.map m (⨅ i, ⨅ (_ : p i), f i) = ⨅ i, ⨅ (_ : p i), Filter.map m (f i)
• 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.HasAntitoneBasis.map 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {β : Type u_2} {ι'' : Type u_6} [Preorder ι''] {l : Filter α} {s : ι'' → Set α} (hf : l.HasAntitoneBasis s) (m : α → β) : (Filter.map m l).HasAntitoneBasis fun x => m '' s x
• Filter.HasBasis.map 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (f : α → β) (hl : l.HasBasis p s) : (Filter.map f l).HasBasis p fun i => f '' s i
• Filter.map_sigma_mk_comap 📋 Mathlib.Order.Filter.Bases.Basic
  {α : Type u_1} {β : Type u_2} {π : α → Type u_6} {π' : β → Type u_7} {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_map 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {f : α → β} {x : Filter α} : Filter.Tendsto f x (Filter.map f x)
• Filter.tendsto_map' 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : β → γ} {g : α → β} {x : Filter α} {y : Filter γ} : Filter.Tendsto (f ∘ g) x y → Filter.Tendsto f (Filter.map g x) y
• Filter.tendsto_map'_iff 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : β → γ} {g : α → β} {x : Filter α} {y : Filter γ} : Filter.Tendsto f (Filter.map g x) y ↔ Filter.Tendsto (f ∘ g) x y
• Filter.map_inf_principal_preimage 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {f : α → β} {s : Set β} {l : Filter α} : Filter.map f (l ⊓ Filter.principal (f ⁻¹' s)) = Filter.map f l ⊓ Filter.principal s
• Filter.Tendsto.map_mapsTo_Iic 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {F : Filter α} {G : Filter β} {m : α → β} : Filter.Tendsto m F G → Set.MapsTo (Filter.map m) (Set.Iic F) (Set.Iic G)
• Filter.map_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 : φ ∘ ψ = id) (hφ : Filter.Tendsto φ f g) (hψ : Filter.Tendsto ψ g f) : Filter.map φ f = g
• Filter.map_mapsTo_Iic_iff_tendsto 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {F : Filter α} {G : Filter β} {m : α → β} : Set.MapsTo (Filter.map m) (Set.Iic F) (Set.Iic G) ↔ Filter.Tendsto m F G
• Filter.le_map_of_right_inverse 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {mab : α → β} {mba : β → α} {f : Filter α} {g : Filter β} (h₁ : mab ∘ mba =ᶠ[g] id) (h₂ : Filter.Tendsto mba g f) : g ≤ Filter.map mab f
• Set.MapsTo.filter_map_Iic 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {m : α → β} : Set.MapsTo m s t → Set.MapsTo (Filter.map m) (Set.Iic (Filter.principal s)) (Set.Iic (Filter.principal t))
• Filter.map_mapsTo_Iic_iff_mapsTo 📋 Mathlib.Order.Filter.Tendsto
  {α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} {m : α → β} : Set.MapsTo (Filter.map m) (Set.Iic (Filter.principal s)) (Set.Iic (Filter.principal t)) ↔ Set.MapsTo m s t
• Nat.map_cast_int_atTop 📋 Mathlib.Order.Filter.AtTopBot.Basic
  : Filter.map Nat.cast Filter.atTop = Filter.atTop
• Filter.map_add_atTop_eq_nat 📋 Mathlib.Order.Filter.AtTopBot.Basic
  (k : ℕ) : Filter.map (fun a => a + k) Filter.atTop = Filter.atTop
• Filter.map_sub_atTop_eq_nat 📋 Mathlib.Order.Filter.AtTopBot.Basic
  (k : ℕ) : Filter.map (fun a => a - k) Filter.atTop = Filter.atTop
• Filter.map_div_atTop_eq_nat 📋 Mathlib.Order.Filter.AtTopBot.Basic
  (k : ℕ) (hk : 0 < k) : Filter.map (fun a => a / k) Filter.atTop = Filter.atTop
• Filter.map_atBot_eq 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} {β : Type u_4} [Preorder α] [IsCodirectedOrder α] [Nonempty α] {f : α → β} : Filter.map f Filter.atBot = ⨅ a, Filter.principal (f '' {a' | a' ≤ a})
• Filter.map_atTop_eq 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirectedOrder α] [Nonempty α] {f : α → β} : Filter.map f Filter.atTop = ⨅ a, Filter.principal (f '' {a' | a ≤ a'})
• Filter.inf_map_atBot_neBot_iff 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} {β : Type u_4} [Preorder α] [IsCodirectedOrder α] {F : Filter β} {u : α → β} [Nonempty α] : (F ⊓ Filter.map u Filter.atBot).NeBot ↔ ∀ U ∈ F, ∀ (N : α), ∃ n ≤ N, u n ∈ U
• Filter.inf_map_atTop_neBot_iff 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirectedOrder α] {F : Filter β} {u : α → β} [Nonempty α] : (F ⊓ Filter.map u Filter.atTop).NeBot ↔ ∀ U ∈ F, ∀ (N : α), ∃ n, N ≤ n ∧ u n ∈ U
• Filter.map_atBot_eq_of_gc_preorder 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} {β : Type u_4} [Preorder α] [IsCodirectedOrder α] [Preorder β] [IsCodirectedOrder β] {f : α → β} (hf : Monotone f) (b : β) (hgi : ∀ c ≤ b, ∃ x, f x = c ∧ ∀ (a : α), c ≤ f a ↔ x ≤ a) : Filter.map f Filter.atBot = Filter.atBot
• Filter.map_atTop_eq_of_gc_preorder 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirectedOrder α] [Preorder β] [IsDirectedOrder β] {f : α → β} (hf : Monotone f) (b : β) (hgi : ∀ (c : β), b ≤ c → ∃ x, f x = c ∧ ∀ (a : α), f a ≤ c ↔ a ≤ x) : Filter.map f Filter.atTop = Filter.atTop
• Filter.map_val_Ici_atTop 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} [Preorder α] [IsDirectedOrder α] (a : α) : Filter.map Subtype.val Filter.atTop = Filter.atTop
• Filter.map_val_Iic_atBot 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} [Preorder α] [IsCodirectedOrder α] (a : α) : Filter.map Subtype.val Filter.atBot = Filter.atBot
• Filter.map_val_atBot_of_Iic_subset 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} [Preorder α] [IsCodirectedOrder α] {a : α} {s : Set α} (h : Set.Iic a ⊆ s) : Filter.map Subtype.val Filter.atBot = Filter.atBot
• Filter.map_val_atTop_of_Ici_subset 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} [Preorder α] [IsDirectedOrder α] {a : α} {s : Set α} (h : Set.Ici a ⊆ s) : Filter.map Subtype.val Filter.atTop = Filter.atTop
• Filter.map_val_Iio_atBot 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} [Preorder α] [IsCodirectedOrder α] [NoMinOrder α] (a : α) : Filter.map Subtype.val Filter.atBot = Filter.atBot
• Filter.map_val_Ioi_atTop 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} [Preorder α] [IsDirectedOrder α] [NoMaxOrder α] (a : α) : Filter.map Subtype.val Filter.atTop = Filter.atTop
• Filter.map_atBot_eq_of_gc 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} {β : Type u_4} [Preorder α] [IsCodirectedOrder α] [PartialOrder β] [IsCodirectedOrder β] {f : α → β} (g : β → α) (b : β) (hf : Monotone f) (gc : ∀ (a : α), ∀ c ≤ b, c ≤ f a ↔ g c ≤ a) (hgi : ∀ c ≤ b, f (g c) ≤ c) : Filter.map f Filter.atBot = Filter.atBot
• Filter.map_atTop_eq_of_gc 📋 Mathlib.Order.Filter.AtTopBot.Basic
  {α : Type u_3} {β : Type u_4} [Preorder α] [IsDirectedOrder α] [PartialOrder β] [IsDirectedOrder β] {f : α → β} (g : β → α) (b : β) (hf : Monotone f) (gc : ∀ (a : α) (c : β), b ≤ c → (f a ≤ c ↔ a ≤ g c)) (hgi : ∀ (c : β), b ≤ c → c ≤ f (g c)) : Filter.map f Filter.atTop = Filter.atTop
• Filter.map_fst_prod 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} (f : Filter α) (g : Filter β) [g.NeBot] : Filter.map Prod.fst (f ×ˢ g) = f
• Filter.map_snd_prod 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} (f : Filter α) (g : Filter β) [f.NeBot] : Filter.map Prod.snd (f ×ˢ g) = g
• Filter.pure_prod 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {a : α} {f : Filter β} : pure a ×ˢ f = Filter.map (Prod.mk a) f
• Filter.prod_pure 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {f : Filter α} {b : β} : f ×ˢ pure b = Filter.map (fun a => (a, b)) f
• Filter.prod_eq 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {f : Filter α} {g : Filter β} : f ×ˢ g = (Filter.map Prod.mk f).seq g
• Filter.map_pure_prod 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) (a : α) (B : Filter β) : Filter.map (Function.uncurry f) (pure a ×ˢ B) = Filter.map (f a) B
• 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.map Prod.swap (g ×ˢ f)
• Filter.map_prod 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} (m : α × β → γ) (f : Filter α) (g : Filter β) : Filter.map m (f ×ˢ g) = (Filter.map (fun a b => m (a, b)) f).seq g
• Filter.le_prod_map_fst_snd 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {f : Filter (α × β)} : f ≤ Filter.map Prod.fst f ×ˢ Filter.map Prod.snd f
• Filter.prod_map_left 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β) (F : Filter α) (G : Filter γ) : Filter.map f F ×ˢ G = Filter.map (Prod.map f id) (F ×ˢ G)
• Filter.prod_map_right 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : β → γ) (F : Filter α) (G : Filter β) : F ×ˢ Filter.map f G = Filter.map (Prod.map id f) (F ×ˢ G)
• Filter.map_prodMap_coprod_le 📋 Mathlib.Order.Filter.Prod
  {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x} {f₁ : Filter α₁} {f₂ : Filter α₂} {m₁ : α₁ → β₁} {m₂ : α₂ → β₂} : Filter.map (Prod.map m₁ m₂) (f₁.coprod f₂) ≤ (Filter.map m₁ f₁).coprod (Filter.map m₂ f₂)
• Filter.prod_map_map_eq' 📋 Mathlib.Order.Filter.Prod
  {α₁ : Type u_6} {α₂ : Type u_7} {β₁ : Type u_8} {β₂ : Type u_9} (f : α₁ → α₂) (g : β₁ → β₂) (F : Filter α₁) (G : Filter β₁) : Filter.map f F ×ˢ Filter.map g G = Filter.map (Prod.map f g) (F ×ˢ G)
• Filter.map_prodMap_const_id_principal_coprod_principal 📋 Mathlib.Order.Filter.Prod
  {α : Type u_6} {β : Type u_7} {ι : Type u_8} (a : α) (b : β) (i : ι) : Filter.map (Prod.map (fun x => b) id) ((Filter.principal {a}).coprod (Filter.principal {i})) = Filter.principal ({b} ×ˢ Set.univ)
• Filter.prod_map_map_eq 📋 Mathlib.Order.Filter.Prod
  {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x} {f₁ : Filter α₁} {f₂ : Filter α₂} {m₁ : α₁ → β₁} {m₂ : α₂ → β₂} : Filter.map m₁ f₁ ×ˢ Filter.map m₂ f₂ = Filter.map (fun p => (m₁ p.1, m₂ p.2)) (f₁ ×ˢ f₂)
• Filter.map_const_principal_coprod_map_id_principal 📋 Mathlib.Order.Filter.Prod
  {α : Type u_6} {β : Type u_7} {ι : Type u_8} (a : α) (b : β) (i : ι) : (Filter.map (fun x => b) (Filter.principal {a})).coprod (Filter.map id (Filter.principal {i})) = Filter.principal ({b} ×ˢ Set.univ ∪ Set.univ ×ˢ {i})
• Filter.prod_assoc 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : Filter α) (g : Filter β) (h : Filter γ) : Filter.map (⇑(Equiv.prodAssoc α β γ)) ((f ×ˢ g) ×ˢ h) = f ×ˢ g ×ˢ h
• Filter.prod_assoc_symm 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : Filter α) (g : Filter β) (h : Filter γ) : Filter.map (⇑(Equiv.prodAssoc α β γ).symm) (f ×ˢ g ×ˢ h) = (f ×ˢ g) ×ˢ h
• Filter.map_swap4_prod 📋 Mathlib.Order.Filter.Prod
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f : Filter α} {g : Filter β} {h : Filter γ} {k : Filter δ} : Filter.map (fun p => ((p.1.1, p.2.1), p.1.2, p.2.2)) ((f ×ˢ g) ×ˢ h ×ˢ k) = (f ×ˢ h) ×ˢ g ×ˢ k
• Filter.lift'_map_le 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : Filter α} {g : Set β → Set γ} {m : α → β} : (Filter.map m f).lift' g ≤ f.lift' (g ∘ Set.image m)
• Filter.lift_map_le 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : Filter α} {g : Set β → Filter γ} {m : α → β} : (Filter.map m f).lift g ≤ f.lift (g ∘ Set.image m)
• Filter.map_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 : β → γ} (hg : Monotone g) : Filter.map m (f.lift g) = f.lift (Filter.map m ∘ g)
• Filter.map_lift_eq2 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : Filter α} {g : Set β → Filter γ} {m : α → β} (hg : Monotone g) : (Filter.map m f).lift g = f.lift (g ∘ Set.image m)
• Filter.map_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 : β → γ} (hh : Monotone h) : Filter.map m (f.lift' h) = f.lift' (Set.image m ∘ h)
• Filter.map_lift'_eq2 📋 Mathlib.Order.Filter.Lift
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : Filter α} {g : Set β → Set γ} {m : α → β} (hg : Monotone g) : (Filter.map m f).lift' g = f.lift' (g ∘ Set.image m)
• map_nhds 📋 Mathlib.Topology.Neighborhoods
  {X : Type u} [TopologicalSpace X] {α : Type u_1} {x : X} {f : X → α} : Filter.map f (nhds x) = ⨅ s ∈ {s | x ∈ s ∧ IsOpen s}, Filter.principal (f '' s)
• MapClusterPt.clusterPt 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {α : Type u_1} {F : Filter α} {u : α → X} {x : X} : MapClusterPt x F u → ClusterPt x (Filter.map u F)
• mapClusterPt_def 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {α : Type u_1} {F : Filter α} {u : α → X} {x : X} : MapClusterPt x F u ↔ ClusterPt x (Filter.map u F)
• mapClusterPt_comp 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {α : Type u_1} {β : Type u_2} {F : Filter α} {x : X} {φ : α → β} {u : β → X} : MapClusterPt x F (u ∘ φ) ↔ MapClusterPt x (Filter.map φ F) u
• MapClusterPt.tendsto_comp' 📋 Mathlib.Topology.ClusterPt
  {X : Type u} [TopologicalSpace X] {Y : Type v} {α : Type u_1} {F : Filter α} {u : α → X} {x : X} [TopologicalSpace Y] {f : X → Y} {y : Y} (hf : Filter.Tendsto f (nhds x ⊓ Filter.map u F) (nhds y)) (hu : MapClusterPt x F u) : MapClusterPt y F (f ∘ u)
• coinduced_nhdsAdjoint 📋 Mathlib.Topology.Order
  {α : Type u} {β : Type v} (f : α → β) (a : α) (l : Filter α) : TopologicalSpace.coinduced f (nhdsAdjoint a l) = nhdsAdjoint (f a) (Filter.map f l)
• map_nhds_induced_eq 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {f : α → β} (a : α) : Filter.map f (nhds a) = nhdsWithin (f a) (Set.range f)
• map_nhds_induced_of_surjective 📋 Mathlib.Topology.Order
  {α : Type u} {β : Type v} [T : TopologicalSpace α] {f : β → α} (hf : Function.Surjective f) (a : β) : Filter.map f (nhds a) = nhds (f a)
• map_nhds_induced_of_mem 📋 Mathlib.Topology.Order
  {α : Type u_1} {β : Type u_2} [t : TopologicalSpace β] {f : α → β} {a : α} (h : Set.range f ∈ nhds (f a)) : Filter.map f (nhds a) = nhds (f a)
• Topology.IsOpenEmbedding.map_nhds_eq 📋 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) : Filter.map f (nhds x) = nhds (f x)
• Topology.IsEmbedding.map_nhds_eq 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Topology.IsEmbedding f) (x : X) : Filter.map f (nhds x) = nhdsWithin (f x) (Set.range f)
• Topology.IsInducing.map_nhds_eq 📋 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) : Filter.map f (nhds x) = nhdsWithin (f x) (Set.range f)
• IsOpenMap.map_nhds_eq 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) {x : X} (hf' : ContinuousAt f x) : Filter.map f (nhds x) = nhds (f x)
• IsOpenMap.nhds_le 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) (x : X) : nhds (f x) ≤ Filter.map f (nhds x)
• IsOpenMap.of_nhds_le 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : ∀ (x : X), nhds (f x) ≤ Filter.map f (nhds x)) : IsOpenMap f
• isOpenMap_iff_nhds_le 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsOpenMap f ↔ ∀ (x : X), nhds (f x) ≤ Filter.map f (nhds x)
• IsOpenMap.map_nhdsSet_eq 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) (hf' : Continuous f) (s : Set X) : Filter.map f (nhdsSet s) = nhdsSet (f '' s)
• IsClosedMap.lift'_closure_map_eq 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (f_closed : IsClosedMap f) (f_cont : Continuous f) (F : Filter X) : (Filter.map f F).lift' closure = Filter.map f (F.lift' closure)
• Topology.IsEmbedding.map_nhds_of_mem 📋 Mathlib.Topology.Maps.Basic
  {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Topology.IsEmbedding f) (x : X) (h : Set.range f ∈ nhds (f x)) : Filter.map f (nhds x) = nhds (f x)
• Topology.IsInducing.map_nhds_of_mem 📋 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) (h : Set.range f ∈ nhds (f x)) : Filter.map f (nhds x) = nhds (f x)
• Homeomorph.map_nhds_eq 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (x : X) : Filter.map (⇑h) (nhds x) = nhds (h x)
• Homeomorph.symm_map_nhds_eq 📋 Mathlib.Topology.Homeomorph.Defs
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (x : X) : Filter.map (⇑h.symm) (nhds (h x)) = nhds x
• map_fst_nhds 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (x : X × Y) : Filter.map Prod.fst (nhds x) = nhds x.1
• map_snd_nhds 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (x : X × Y) : Filter.map Prod.snd (nhds x) = nhds x.2
• nhds_inl 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (x : X) : nhds (Sum.inl x) = Filter.map Sum.inl (nhds x)
• nhds_inr 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (y : Y) : nhds (Sum.inr y) = Filter.map Sum.inr (nhds y)
• nhds_swap 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (x : X) (y : Y) : nhds (x, y) = Filter.map Prod.swap (nhds (y, x))
• map_fst_nhdsWithin 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (x : X × Y) : Filter.map Prod.fst (nhdsWithin x (Prod.snd ⁻¹' {x.2})) = nhds x.1
• map_snd_nhdsWithin 📋 Mathlib.Topology.Constructions.SumProd
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] (x : X × Y) : Filter.map Prod.snd (nhdsWithin x (Prod.fst ⁻¹' {x.1})) = nhds x.2
• uniformity_eq_symm 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] : uniformity α = Filter.map Prod.swap (uniformity α)
• symm_le_uniformity 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] : Filter.map Prod.swap (uniformity α) ≤ uniformity α
• uniformity_le_symm 📋 Mathlib.Topology.UniformSpace.Defs
  {α : Type ua} [UniformSpace α] : uniformity α ≤ Filter.map Prod.swap (uniformity α)
• Filter.map.isCountablyGenerated 📋 Mathlib.Order.Filter.CountablyGenerated
  {α : Type u_1} {β : Type u_2} (l : Filter α) [l.IsCountablyGenerated] (f : α → β) : (Filter.map f l).IsCountablyGenerated
• Filter.map_eval_pi 📋 Mathlib.Order.Filter.Pi
  {ι : Type u_1} {α : ι → Type u_2} (f : (i : ι) → Filter (α i)) [∀ (i : ι), (f i).NeBot] (i : ι) : Filter.map (Function.eval i) (Filter.pi f) = f i
• Filter.map_pi_map_coprodᵢ_le 📋 Mathlib.Order.Filter.Pi
  {ι : Type u_1} {α : ι → Type u_2} {f : (i : ι) → Filter (α i)} {β : ι → Type u_3} {m : (i : ι) → α i → β i} : Filter.map (fun k i => m i (k i)) (Filter.coprodᵢ f) ≤ Filter.coprodᵢ fun i => Filter.map (m i) (f i)
• Function.Surjective.le_map_cofinite 📋 Mathlib.Order.Filter.Cofinite
  {α : Type u_2} {β : Type u_3} {f : α → β} (hf : Function.Surjective f) : Filter.cofinite ≤ Filter.map f Filter.cofinite
• Filter.map_piMap_pi_finite 📋 Mathlib.Order.Filter.Cofinite
  {ι : Type u_1} {α : ι → Type u_4} {β : ι → Type u_5} [Finite ι] (f : (i : ι) → α i → β i) (l : (i : ι) → Filter (α i)) : Filter.map (Pi.map f) (Filter.pi l) = Filter.pi fun i => Filter.map (f i) (l i)
• Filter.map_piMap_pi 📋 Mathlib.Order.Filter.Cofinite
  {ι : Type u_1} {α : ι → Type u_4} {β : ι → Type u_5} {f : (i : ι) → α i → β i} (hf : ∀ᶠ (i : ι) in Filter.cofinite, Function.Surjective (f i)) (l : (i : ι) → Filter (α i)) : Filter.map (Pi.map f) (Filter.pi l) = Filter.pi fun i => Filter.map (f i) (l i)
• Sigma.nhds_mk 📋 Mathlib.Topology.Constructions
  {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] (i : ι) (x : σ i) : nhds ⟨i, x⟩ = Filter.map (Sigma.mk i) (nhds x)
• map_nhds_subtype_coe_eq_nhds 📋 Mathlib.Topology.Constructions
  {X : Type u} [TopologicalSpace X] {p : X → Prop} {x : X} (hx : p x) (h : ∀ᶠ (x : X) in nhds x, p x) : Filter.map Subtype.val (nhds ⟨x, hx⟩) = nhds x
• Sigma.nhds_eq 📋 Mathlib.Topology.Constructions
  {ι : Type u_5} {σ : ι → Type u_7} [(i : ι) → TopologicalSpace (σ i)] (x : Sigma σ) : nhds x = Filter.map (Sigma.mk x.fst) (nhds x.snd)
• nhds_ofAdd 📋 Mathlib.Topology.Constructions
  {X : Type u} [TopologicalSpace X] (x : X) : nhds (Multiplicative.ofAdd x) = Filter.map (⇑Multiplicative.ofAdd) (nhds x)
• nhds_ofMul 📋 Mathlib.Topology.Constructions
  {X : Type u} [TopologicalSpace X] (x : X) : nhds (Additive.ofMul x) = Filter.map (⇑Additive.ofMul) (nhds x)
• nhds_toDual 📋 Mathlib.Topology.Constructions
  {X : Type u} [TopologicalSpace X] (x : X) : nhds (OrderDual.toDual x) = Filter.map (⇑OrderDual.toDual) (nhds x)
• nhds_ofDual 📋 Mathlib.Topology.Constructions
  {X : Type u} [TopologicalSpace X] (x : X) : nhds (OrderDual.ofDual x) = Filter.map (⇑OrderDual.ofDual) (nhds x)
• nhds_toAdd 📋 Mathlib.Topology.Constructions
  {X : Type u} [TopologicalSpace X] (x : Multiplicative X) : nhds (Multiplicative.toAdd x) = Filter.map (⇑Multiplicative.toAdd) (nhds x)
• nhds_toMul 📋 Mathlib.Topology.Constructions
  {X : Type u} [TopologicalSpace X] (x : Additive X) : nhds (Additive.toMul x) = Filter.map (⇑Additive.toMul) (nhds x)

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 a114d38 serving mathlib revision 0d14bcb