Sets in sigma types #

This file defines Set.sigma, the indexed sum of sets.

@[simp]

theorem Set.range_sigmaMk {ι : Type u_1} {α : ι → Type u_2} (i : ι) :

Set.range (Sigma.mk i) = Sigma.fst ⁻¹' {i}

theorem Set.preimage_image_sigmaMk_of_ne {ι : Type u_1} {α : ι → Type u_2} {i : ι} {j : ι} (h : i ≠ j) (s : Set (α j)) :

Sigma.mk i ⁻¹' (Sigma.mk j '' s) = ∅

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theorem Set.image_sigmaMk_preimage_sigmaMap_subset {ι : Type u_2} {ι' : Type u_4} {α : ι → Type u_3} {β : ι' → Type u_1} (f : ι → ι') (g : (i : ι) → α i → β (f i)) (i : ι) (s : Set (β (f i))) :

Sigma.mk i '' (g i ⁻¹' s) ⊆ Sigma.map f g ⁻¹' (Sigma.mk (f i) '' s)

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theorem Set.image_sigmaMk_preimage_sigmaMap {ι : Type u_2} {ι' : Type u_3} {α : ι → Type u_4} {β : ι' → Type u_1} {f : ι → ι'} (hf : Function.Injective f) (g : (i : ι) → α i → β (f i)) (i : ι) (s : Set (β (f i))) :

Sigma.mk i '' (g i ⁻¹' s) = Sigma.map f g ⁻¹' (Sigma.mk (f i) '' s)

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def Set.Sigma {ι : Type u_1} {α : ι → Type u_2} (s : Set ι) (t : (i : ι) → Set (α i)) :

Set ((i : ι) × α i)

Indexed sum of sets. s.sigma t is the set of dependent pairs ⟨i, a⟩ such that i ∈ s∈ s and a ∈ t i∈ t i.

Equations

Set.Sigma s t = { x | x.fst ∈ s ∧ x.snd ∈ t x.fst }

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@[simp]

theorem Set.mem_sigma_iff {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {x : (i : ι) × α i} :

x ∈ Set.Sigma s t ↔ x.fst ∈ s ∧ x.snd ∈ t x.fst

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theorem Set.mk_sigma_iff {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {i : ι} {a : α i} :

{ fst := i, snd := a } ∈ Set.Sigma s t ↔ i ∈ s ∧ a ∈ t i

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theorem Set.mk_mem_sigma {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {i : ι} {a : α i} (hi : i ∈ s) (ha : a ∈ t i) :

{ fst := i, snd := a } ∈ Set.Sigma s t

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theorem Set.sigma_mono {ι : Type u_1} {α : ι → Type u_2} {s₁ : Set ι} {s₂ : Set ι} {t₁ : (i : ι) → Set (α i)} {t₂ : (i : ι) → Set (α i)} (hs : s₁ ⊆ s₂) (ht : ∀ (i : ι), t₁ i ⊆ t₂ i) :

Set.Sigma s₁ t₁ ⊆ Set.Sigma s₂ t₂

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theorem Set.sigma_subset_iff {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {u : Set ((i : ι) × α i)} :

Set.Sigma s t ⊆ u ↔ ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃a : α i⦄, a ∈ t i → { fst := i, snd := a } ∈ u

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theorem Set.forall_sigma_iff {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {p : (i : ι) × α i → Prop} :

((x : (i : ι) × α i) → x ∈ Set.Sigma s t → p x) ↔ ⦃i : ι⦄ → i ∈ s → ⦃a : α i⦄ → a ∈ t i → p { fst := i, snd := a }

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theorem Set.exists_sigma_iff {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {p : (i : ι) × α i → Prop} :

(∃ x, x ∈ Set.Sigma s t ∧ p x) ↔ ∃ i, i ∈ s ∧ ∃ a, a ∈ t i ∧ p { fst := i, snd := a }

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theorem Set.sigma_empty {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} :

(Set.Sigma s fun i => ∅) = ∅

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theorem Set.empty_sigma {ι : Type u_1} {α : ι → Type u_2} {t : (i : ι) → Set (α i)} :

Set.Sigma ∅ t = ∅

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theorem Set.univ_sigma_univ {ι : Type u_1} {α : ι → Type u_2} {i : ι} :

(Set.Sigma Set.univ fun x => Set.univ) = Set.univ

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@[simp]

theorem Set.sigma_univ {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} :

(Set.Sigma s fun x => Set.univ) = Sigma.fst ⁻¹' s

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@[simp]

theorem Set.singleton_sigma {ι : Type u_1} {α : ι → Type u_2} {t : (i : ι) → Set (α i)} {i : ι} {a : α i} :

Set.Sigma {i} t = Sigma.mk i '' t i

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theorem Set.sigma_singleton {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {a : (i : ι) → α i} :

(Set.Sigma s fun i => {a i}) = (fun i => { fst := i, snd := a i }) '' s

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theorem Set.singleton_sigma_singleton {ι : Type u_1} {α : ι → Type u_2} {i : ι} {a : (i : ι) → α i} :

(Set.Sigma {i} fun i => {a i}) = {{ fst := i, snd := a i }}

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theorem Set.union_sigma {ι : Type u_1} {α : ι → Type u_2} {s₁ : Set ι} {s₂ : Set ι} {t : (i : ι) → Set (α i)} :

Set.Sigma (s₁ ∪ s₂) t = Set.Sigma s₁ t ∪ Set.Sigma s₂ t

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theorem Set.sigma_union {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t₁ : (i : ι) → Set (α i)} {t₂ : (i : ι) → Set (α i)} :

(Set.Sigma s fun i => t₁ i ∪ t₂ i) = Set.Sigma s t₁ ∪ Set.Sigma s t₂

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theorem Set.sigma_inter_sigma {ι : Type u_1} {α : ι → Type u_2} {s₁ : Set ι} {s₂ : Set ι} {t₁ : (i : ι) → Set (α i)} {t₂ : (i : ι) → Set (α i)} :

Set.Sigma s₁ t₁ ∩ Set.Sigma s₂ t₂ = Set.Sigma (s₁ ∩ s₂) fun i => t₁ i ∩ t₂ i

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theorem Set.insert_sigma {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {i : ι} {a : α i} :

Set.Sigma (insert i s) t = Sigma.mk i '' t i ∪ Set.Sigma s t

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theorem Set.sigma_insert {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {a : (i : ι) → α i} :

(Set.Sigma s fun i => insert (a i) (t i)) = (fun i => { fst := i, snd := a i }) '' s ∪ Set.Sigma s t

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theorem Set.sigma_preimage_eq {ι : Type u_3} {ι' : Type u_1} {α : ι → Type u_4} {β : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {f : ι' → ι} {g : (i : ι) → β i → α i} :

(Set.Sigma (f ⁻¹' s) fun i => g (f i) ⁻¹' t (f i)) = (fun p => { fst := f p.fst, snd := g (f p.fst) p.snd }) ⁻¹' Set.Sigma s t

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theorem Set.sigma_preimage_left {ι : Type u_3} {ι' : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {f : ι' → ι} :

(Set.Sigma (f ⁻¹' s) fun i => t (f i)) = (fun p => { fst := f p.fst, snd := p.snd }) ⁻¹' Set.Sigma s t

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theorem Set.sigma_preimage_right {ι : Type u_1} {α : ι → Type u_3} {β : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {g : (i : ι) → β i → α i} :

(Set.Sigma s fun i => g i ⁻¹' t i) = (fun p => { fst := p.fst, snd := g p.fst p.snd }) ⁻¹' Set.Sigma s t

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theorem Set.preimage_sigmaMap_sigma {ι : Type u_3} {ι' : Type u_2} {α : ι → Type u_4} {α' : ι' → Type u_1} (f : ι → ι') (g : (i : ι) → α i → α' (f i)) (s : Set ι') (t : (i : ι') → Set (α' i)) :

Sigma.map f g ⁻¹' Set.Sigma s t = Set.Sigma (f ⁻¹' s) fun i => g i ⁻¹' t (f i)

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@[simp]

theorem Set.mk_preimage_sigma {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {i : ι} (hi : i ∈ s) :

Sigma.mk i ⁻¹' Set.Sigma s t = t i

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theorem Set.mk_preimage_sigma_eq_empty {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {i : ι} (hi : ¬i ∈ s) :

Sigma.mk i ⁻¹' Set.Sigma s t = ∅

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theorem Set.mk_preimage_sigma_eq_if {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {i : ι} [inst : DecidablePred fun x => x ∈ s] :

Sigma.mk i ⁻¹' Set.Sigma s t = if i ∈ s then t i else ∅

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theorem Set.mk_preimage_sigma_fn_eq_if {ι : Type u_2} {α : ι → Type u_3} {s : Set ι} {t : (i : ι) → Set (α i)} {i : ι} {β : Type u_1} [inst : DecidablePred fun x => x ∈ s] (g : β → α i) :

(fun b => { fst := i, snd := g b }) ⁻¹' Set.Sigma s t = if i ∈ s then g ⁻¹' t i else ∅

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theorem Set.sigma_univ_range_eq {ι : Type u_1} {α : ι → Type u_3} {β : ι → Type u_2} {f : (i : ι) → α i → β i} :

(Set.Sigma Set.univ fun i => Set.range (f i)) = Set.range fun x => { fst := x.fst, snd := f x.fst x.snd }

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theorem Set.Nonempty.sigma {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} :

Set.Nonempty s → (∀ (i : ι), Set.Nonempty (t i)) → Set.Nonempty (Set.Sigma s t)

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theorem Set.Nonempty.sigma_fst {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} :

Set.Nonempty (Set.Sigma s t) → Set.Nonempty s

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theorem Set.Nonempty.sigma_snd {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} :

Set.Nonempty (Set.Sigma s t) → ∃ i, i ∈ s ∧ Set.Nonempty (t i)

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theorem Set.sigma_nonempty_iff {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} :

Set.Nonempty (Set.Sigma s t) ↔ ∃ i, i ∈ s ∧ Set.Nonempty (t i)

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theorem Set.sigma_eq_empty_iff {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} :

Set.Sigma s t = ∅ ↔ ∀ (i : ι), i ∈ s → t i = ∅

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theorem Set.image_sigmaMk_subset_sigma_left {ι : Type u_2} {α : ι → Type u_1} {s : Set ι} {t : (i : ι) → Set (α i)} {a : (i : ι) → α i} (ha : ∀ (i : ι), a i ∈ t i) :

(fun i => { fst := i, snd := a i }) '' s ⊆ Set.Sigma s t

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theorem Set.image_sigmaMk_subset_sigma_right {ι : Type u_1} {α : ι → Type u_2} {s : Set ι} {t : (i : ι) → Set (α i)} {i : ι} (hi : i ∈ s) :

Sigma.mk i '' t i ⊆ Set.Sigma s t

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theorem Set.sigma_subset_preimage_fst {ι : Type u_1} {α : ι → Type u_2} (s : Set ι) (t : (i : ι) → Set (α i)) :

Set.Sigma s t ⊆ Sigma.fst ⁻¹' s

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theorem Set.fst_image_sigma_subset {ι : Type u_1} {α : ι → Type u_2} (s : Set ι) (t : (i : ι) → Set (α i)) :

Sigma.fst '' Set.Sigma s t ⊆ s

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theorem Set.fst_image_sigma {ι : Type u_1} {α : ι → Type u_2} {t : (i : ι) → Set (α i)} (s : Set ι) (ht : ∀ (i : ι), Set.Nonempty (t i)) :

Sigma.fst '' Set.Sigma s t = s

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theorem Set.sigma_diff_sigma {ι : Type u_1} {α : ι → Type u_2} {s₁ : Set ι} {s₂ : Set ι} {t₁ : (i : ι) → Set (α i)} {t₂ : (i : ι) → Set (α i)} :

Set.Sigma s₁ t₁ \ Set.Sigma s₂ t₂ = Set.Sigma s₁ (t₁ \ t₂) ∪ Set.Sigma (s₁ \ s₂) t₁