File: Mathlib/Data/Set/Subset.lean

full_name: Set.preimage_val_eq_univ_of_subset
code: lemma preimage_val_eq_univ_of_subset (h : A ⊆ B) : A ↓∩ B = univ 
goal: Subtype.val ⁻¹' B = Set.univ

full_name: Set.preimage_val_sUnion
code: lemma preimage_val_sUnion : A ↓∩ (⋃₀ S) = ⋃₀ { (A ↓∩ B) | B ∈ S } 
goal: Subtype.val ⁻¹' ⋃₀ S = ⋃₀ {x | ∃ B ∈ S, Subtype.val ⁻¹' B = x}

full_name: Set.preimage_val_iInter
code: @[simp]
lemma preimage_val_iInter : A ↓∩ (⋂ i, s i) = ⋂ i, A ↓∩ s i 
goal: Subtype.val ⁻¹' ⋂ i, s i = ⋂ i, Subtype.val ⁻¹' s i

full_name: Set.preimage_val_sInter
code: lemma preimage_val_sInter : A ↓∩ (⋂₀ S) = ⋂₀ { (A ↓∩ B) | B ∈ S } 
goal: Subtype.val ⁻¹' ⋂₀ S = ⋂₀ {x | ∃ B ∈ S, Subtype.val ⁻¹' B = x}

full_name: Set.preimage_val_sInter_eq_sInter
code: lemma preimage_val_sInter_eq_sInter : A ↓∩ (⋂₀ S) = ⋂₀ ((A ↓∩ ·) '' S) 
goal: Subtype.val ⁻¹' ⋂₀ S = ⋂₀ ((fun x => Subtype.val ⁻¹' x) '' S)

full_name: Set.eq_of_preimage_val_eq_of_subset
code: lemma eq_of_preimage_val_eq_of_subset (hB : B ⊆ A) (hC : C ⊆ A) (h : A ↓∩ B = A ↓∩ C) : B = C 
goal: B = C

full_name: Set.image_val_union
code: @[simp]
lemma image_val_union : (↑(D ∪ E) : Set α) = ↑D ∪ ↑E 
goal: Subtype.val '' (D ∪ E) = Subtype.val '' D ∪ Subtype.val '' E

full_name: Set.image_val_inter
code: @[simp]
lemma image_val_inter : (↑(D ∩ E) : Set α) = ↑D ∩ ↑E 
goal: Subtype.val '' (D ∩ E) = Subtype.val '' D ∩ Subtype.val '' E

full_name: Set.image_val_diff
code: @[simp]
lemma image_val_diff : (↑(D \ E) : Set α) = ↑D \ ↑E 
goal: Subtype.val '' (D \ E) = Subtype.val '' D \ Subtype.val '' E

full_name: Set.image_val_compl
code: @[simp]
lemma image_val_compl : ↑(Dᶜ) = A \ ↑D 
goal: Subtype.val '' Dᶜ = A \ Subtype.val '' D

full_name: Set.image_val_sUnion
code: @[simp]
lemma image_val_sUnion : ↑(⋃₀ T) = ⋃₀ { (B : Set α) | B ∈ T} 
goal: Subtype.val '' ⋃₀ T = ⋃₀ {x | ∃ B ∈ T, Subtype.val '' B = x}

full_name: Set.image_val_iUnion
code: @[simp]
lemma image_val_iUnion : ↑(⋃ i, t i) = ⋃ i, (t i : Set α) 
goal: Subtype.val '' ⋃ i, t i = ⋃ i, Subtype.val '' t i

full_name: Set.image_val_sInter
code: @[simp]
lemma image_val_sInter (hT : T.Nonempty) : (↑(⋂₀ T) : Set α) = ⋂₀ { (↑B : Set α) | B ∈ T } 
goal: Subtype.val '' ⋂₀ T = ⋂₀ {x | ∃ B ∈ T, Subtype.val '' B = x}

full_name: Set.image_val_iInter
code: @[simp]
lemma image_val_iInter [Nonempty ι] : (↑(⋂ i, t i) : Set α) = ⋂ i, (↑(t i) : Set α) 
goal: Subtype.val '' ⋂ i, t i = ⋂ i, Subtype.val '' t i

full_name: Set.image_val_union_self_right_eq
code: @[simp]
lemma image_val_union_self_right_eq : A ∪ ↑D = A 
goal: A ∪ Subtype.val '' D = A

full_name: Set.image_val_union_self_left_eq
code: @[simp]
lemma image_val_union_self_left_eq : ↑D ∪ A = A 
goal: Subtype.val '' D ∪ A = A

full_name: Set.cou_inter_self_right_eq_coe
code: @[simp]
lemma cou_inter_self_right_eq_coe : A ∩ ↑D = ↑D 
goal: A ∩ Subtype.val '' D = Subtype.val '' D

full_name: Set.image_val_inter_self_left_eq_coe
code: @[simp]
lemma image_val_inter_self_left_eq_coe : ↑D ∩ A = ↑D 
goal: Subtype.val '' D ∩ A = Subtype.val '' D

full_name: Set.subset_preimage_val_image_val_iff
code: lemma subset_preimage_val_image_val_iff : D ⊆ A ↓∩ ↑E ↔ D ⊆ E 
goal: D ⊆ Subtype.val ⁻¹' (Subtype.val '' E) ↔ D ⊆ E

full_name: Set.image_val_inj
code: @[simp]
lemma image_val_inj : (D : Set α) = ↑E ↔ D = E 
goal: Subtype.val '' D = Subtype.val '' E ↔ D = E

full_name: Set.image_val_injective
code: lemma image_val_injective : Function.Injective ((↑) : Set A → Set α) 
goal: Function.Injective (Set.image Subtype.val)

full_name: Set.subset_of_image_val_subset_image_val
code: lemma subset_of_image_val_subset_image_val (h : (↑D : Set α) ⊆ ↑E) : D ⊆ E 
goal: D ⊆ E

full_name: Set.image_val_mono
code: @[mono]
lemma image_val_mono (h : D ⊆ E) : (↑D : Set α) ⊆ ↑E 
goal: Subtype.val '' D ⊆ Subtype.val '' E

full_name: Set.image_val_preimage_val_subset_self
code: lemma image_val_preimage_val_subset_self : ↑(A ↓∩ B) ⊆ B 
goal: Subtype.val '' (Subtype.val ⁻¹' B) ⊆ B

full_name: Set.preimage_val_image_val_eq_self
code: lemma preimage_val_image_val_eq_self : A ↓∩ ↑D = D 
goal: Subtype.val ⁻¹' (Subtype.val '' D) = D