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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | xrninxp 37801* | Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 7-Apr-2020.) |
⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = ◡{⟨⟨𝑦, 𝑧⟩, 𝑢⟩ ∣ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝑦, 𝑧⟩))} | ||
Theorem | xrninxp2 37802* | Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 8-Apr-2020.) |
⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))} | ||
Theorem | xrninxpex 37803 | Sufficient condition for the intersection of a range Cartesian product with a Cartesian product to be a set. (Contributed by Peter Mazsa, 12-Apr-2020.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V) | ||
Theorem | inxpxrn 37804 | Two ways to express the intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 10-Apr-2020.) |
⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶))) = ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) | ||
Theorem | br1cnvxrn2 37805* | The converse of a binary relation over a range Cartesian product. (Contributed by Peter Mazsa, 11-Jul-2021.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴◡(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))) | ||
Theorem | elec1cnvxrn2 37806* | Elementhood in the converse range Cartesian product coset of 𝐴. (Contributed by Peter Mazsa, 11-Jul-2021.) |
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ [𝐴]◡(𝑅 ⋉ 𝑆) ↔ ∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))) | ||
Theorem | rnxrn 37807* | Range of the range Cartesian product of classes. (Contributed by Peter Mazsa, 1-Jun-2020.) |
⊢ ran (𝑅 ⋉ 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} | ||
Theorem | rnxrnres 37808* | Range of a range Cartesian product with a restricted relation. (Contributed by Peter Mazsa, 5-Dec-2021.) |
⊢ ran (𝑅 ⋉ (𝑆 ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} | ||
Theorem | rnxrncnvepres 37809* | Range of a range Cartesian product with a restriction of the converse epsilon relation. (Contributed by Peter Mazsa, 6-Dec-2021.) |
⊢ ran (𝑅 ⋉ (◡ E ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ 𝑢 ∧ 𝑢𝑅𝑥)} | ||
Theorem | rnxrnidres 37810* | Range of a range Cartesian product with a restriction of the identity relation. (Contributed by Peter Mazsa, 6-Dec-2021.) |
⊢ ran (𝑅 ⋉ ( I ↾ 𝐴)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥)} | ||
Theorem | xrnres 37811 | Two ways to express restriction of range Cartesian product, see also xrnres2 37812, xrnres3 37813. (Contributed by Peter Mazsa, 5-Jun-2021.) |
⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ 𝑆) | ||
Theorem | xrnres2 37812 | Two ways to express restriction of range Cartesian product, see also xrnres 37811, xrnres3 37813. (Contributed by Peter Mazsa, 6-Sep-2021.) |
⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴)) | ||
Theorem | xrnres3 37813 | Two ways to express restriction of range Cartesian product, see also xrnres 37811, xrnres2 37812. (Contributed by Peter Mazsa, 28-Mar-2020.) |
⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) | ||
Theorem | xrnres4 37814 | Two ways to express restriction of range Cartesian product. (Contributed by Peter Mazsa, 29-Dec-2020.) |
⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (ran (𝑅 ↾ 𝐴) × ran (𝑆 ↾ 𝐴)))) | ||
Theorem | xrnresex 37815 | Sufficient condition for a restricted range Cartesian product to be a set. (Contributed by Peter Mazsa, 16-Dec-2020.) (Revised by Peter Mazsa, 7-Sep-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → (𝑅 ⋉ (𝑆 ↾ 𝐴)) ∈ V) | ||
Theorem | xrnidresex 37816 | Sufficient condition for a range Cartesian product with restricted identity to be a set. (Contributed by Peter Mazsa, 31-Dec-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 ⋉ ( I ↾ 𝐴)) ∈ V) | ||
Theorem | xrncnvepresex 37817 | Sufficient condition for a range Cartesian product with restricted converse epsilon to be a set. (Contributed by Peter Mazsa, 16-Dec-2020.) (Revised by Peter Mazsa, 23-Sep-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V) | ||
Theorem | brin2 37818 | Binary relation on an intersection is a special case of binary relation on range Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆)⟨𝐵, 𝐵⟩)) | ||
Theorem | brin3 37819 | Binary relation on an intersection is a special case of binary relation on range Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021.) (Avoid depending on this detail.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆){{𝐵}})) | ||
Definition | df-coss 37820* |
Define the class of cosets by 𝑅: 𝑥 and 𝑦 are cosets by
𝑅 iff there exists a set 𝑢 such
that both 𝑢𝑅𝑥 and
𝑢𝑅𝑦 hold, i.e., both 𝑥 and
𝑦
are are elements of the 𝑅
-coset of 𝑢 (see dfcoss2 37822 and the comment of dfec2 8721). 𝑅 is
usually a relation.
This concept simplifies theorems relating partition and equivalence: the left side of these theorems relate to 𝑅, the right side relate to ≀ 𝑅 (see e.g. pet 38260). Without the definition of ≀ 𝑅 we should have to relate the right side of these theorems to a composition of a converse (cf. dfcoss3 37823) or to the range of a range Cartesian product of classes (cf. dfcoss4 37824), which would make the theorems complicated and confusing. Alternate definition is dfcoss2 37822. Technically, we can define it via composition (dfcoss3 37823) or as the range of a range Cartesian product (dfcoss4 37824), but neither of these definitions reveal directly how the cosets by 𝑅 relate to each other. We define functions (df-funsALTV 38090, df-funALTV 38091) and disjoints (dfdisjs 38117, dfdisjs2 38118, df-disjALTV 38114, dfdisjALTV2 38123) with the help of it as well. (Contributed by Peter Mazsa, 9-Jan-2018.) |
⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | ||
Definition | df-coels 37821 | Define the class of coelements on the class 𝐴, see also the alternate definition dfcoels 37839. Possible definitions are the special cases of dfcoss3 37823 and dfcoss4 37824. (Contributed by Peter Mazsa, 20-Nov-2019.) |
⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | ||
Theorem | dfcoss2 37822* | Alternate definition of the class of cosets by 𝑅: 𝑥 and 𝑦 are cosets by 𝑅 iff there exists a set 𝑢 such that both 𝑥 and 𝑦 are are elements of the 𝑅-coset of 𝑢 (see also the comment of dfec2 8721). 𝑅 is usually a relation. (Contributed by Peter Mazsa, 16-Jan-2018.) |
⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)} | ||
Theorem | dfcoss3 37823 | Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 37820). (Contributed by Peter Mazsa, 27-Dec-2018.) |
⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅) | ||
Theorem | dfcoss4 37824 | Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 37820). (Contributed by Peter Mazsa, 12-Jul-2021.) |
⊢ ≀ 𝑅 = ran (𝑅 ⋉ 𝑅) | ||
Theorem | cosscnv 37825* | Class of cosets by the converse of 𝑅 (Contributed by Peter Mazsa, 17-Jun-2020.) |
⊢ ≀ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)} | ||
Theorem | coss1cnvres 37826* | Class of cosets by the converse of a restriction. (Contributed by Peter Mazsa, 8-Jun-2020.) |
⊢ ≀ ◡(𝑅 ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))} | ||
Theorem | coss2cnvepres 37827* | Special case of coss1cnvres 37826. (Contributed by Peter Mazsa, 8-Jun-2020.) |
⊢ ≀ ◡(◡ E ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣))} | ||
Theorem | cossex 37828 | If 𝐴 is a set then the class of cosets by 𝐴 is a set. (Contributed by Peter Mazsa, 4-Jan-2019.) |
⊢ (𝐴 ∈ 𝑉 → ≀ 𝐴 ∈ V) | ||
Theorem | cosscnvex 37829 | If 𝐴 is a set then the class of cosets by the converse of 𝐴 is a set. (Contributed by Peter Mazsa, 18-Oct-2019.) |
⊢ (𝐴 ∈ 𝑉 → ≀ ◡𝐴 ∈ V) | ||
Theorem | 1cosscnvepresex 37830 | Sufficient condition for a restricted converse epsilon coset to be a set. (Contributed by Peter Mazsa, 24-Sep-2021.) |
⊢ (𝐴 ∈ 𝑉 → ≀ (◡ E ↾ 𝐴) ∈ V) | ||
Theorem | 1cossxrncnvepresex 37831 | Sufficient condition for a restricted converse epsilon range Cartesian product to be a set. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V) | ||
Theorem | relcoss 37832 | Cosets by 𝑅 is a relation. (Contributed by Peter Mazsa, 27-Dec-2018.) |
⊢ Rel ≀ 𝑅 | ||
Theorem | relcoels 37833 | Coelements on 𝐴 is a relation. (Contributed by Peter Mazsa, 5-Oct-2021.) |
⊢ Rel ∼ 𝐴 | ||
Theorem | cossss 37834 | Subclass theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 11-Nov-2019.) |
⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵) | ||
Theorem | cosseq 37835 | Equality theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 9-Jan-2018.) |
⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵) | ||
Theorem | cosseqi 37836 | Equality theorem for the classes of cosets by 𝐴 and 𝐵, inference form. (Contributed by Peter Mazsa, 9-Jan-2018.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ≀ 𝐴 = ≀ 𝐵 | ||
Theorem | cosseqd 37837 | Equality theorem for the classes of cosets by 𝐴 and 𝐵, deduction form. (Contributed by Peter Mazsa, 4-Nov-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ≀ 𝐴 = ≀ 𝐵) | ||
Theorem | 1cossres 37838* | The class of cosets by a restriction. (Contributed by Peter Mazsa, 20-Apr-2019.) |
⊢ ≀ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | ||
Theorem | dfcoels 37839* | Alternate definition of the class of coelements on the class 𝐴. (Contributed by Peter Mazsa, 20-Apr-2019.) |
⊢ ∼ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} | ||
Theorem | brcoss 37840* | 𝐴 and 𝐵 are cosets by 𝑅: a binary relation. (Contributed by Peter Mazsa, 27-Dec-2018.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵))) | ||
Theorem | brcoss2 37841* | Alternate form of the 𝐴 and 𝐵 are cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 26-Mar-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝐴 ∈ [𝑢]𝑅 ∧ 𝐵 ∈ [𝑢]𝑅))) | ||
Theorem | brcoss3 37842 | Alternate form of the 𝐴 and 𝐵 are cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 26-Mar-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ([𝐴]◡𝑅 ∩ [𝐵]◡𝑅) ≠ ∅)) | ||
Theorem | brcosscnvcoss 37843 | For sets, the 𝐴 and 𝐵 cosets by 𝑅 binary relation and the 𝐵 and 𝐴 cosets by 𝑅 binary relation are the same. (Contributed by Peter Mazsa, 27-Dec-2018.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ 𝐵 ≀ 𝑅𝐴)) | ||
Theorem | brcoels 37844* | 𝐵 and 𝐶 are coelements : a binary relation. (Contributed by Peter Mazsa, 14-Jan-2020.) (Revised by Peter Mazsa, 5-Oct-2021.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∼ 𝐴𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢))) | ||
Theorem | cocossss 37845* | Two ways of saying that cosets by cosets by 𝑅 is a subclass. (Contributed by Peter Mazsa, 17-Sep-2021.) |
⊢ ( ≀ ≀ 𝑅 ⊆ 𝑆 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧)) | ||
Theorem | cnvcosseq 37846 | The converse of cosets by 𝑅 are cosets by 𝑅. (Contributed by Peter Mazsa, 3-May-2019.) |
⊢ ◡ ≀ 𝑅 = ≀ 𝑅 | ||
Theorem | br2coss 37847 | Cosets by ≀ 𝑅 binary relation. (Contributed by Peter Mazsa, 25-Aug-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅)) | ||
Theorem | br1cossres 37848* | 𝐵 and 𝐶 are cosets by a restriction: a binary relation. (Contributed by Peter Mazsa, 30-Dec-2018.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ↾ 𝐴)𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑅𝐶))) | ||
Theorem | br1cossres2 37849* | 𝐵 and 𝐶 are cosets by a restriction: a binary relation. (Contributed by Peter Mazsa, 3-Jan-2018.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ↾ 𝐴)𝐶 ↔ ∃𝑥 ∈ 𝐴 (𝐵 ∈ [𝑥]𝑅 ∧ 𝐶 ∈ [𝑥]𝑅))) | ||
Theorem | brressn 37850 | Binary relation on a restriction to a singleton. (Contributed by Peter Mazsa, 11-Jun-2024.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝑅 ↾ {𝐴})𝐶 ↔ (𝐵 = 𝐴 ∧ 𝐵𝑅𝐶))) | ||
Theorem | ressn2 37851* | A class ' R ' restricted to the singleton of the class ' A ' is the ordered pair class abstraction of the class ' A ' and the sets in relation ' R ' to ' A ' (and not in relation to the singleton ' { A } ' ). (Contributed by Peter Mazsa, 16-Jun-2024.) |
⊢ (𝑅 ↾ {𝐴}) = {⟨𝑎, 𝑢⟩ ∣ (𝑎 = 𝐴 ∧ 𝐴𝑅𝑢)} | ||
Theorem | refressn 37852* | Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 37851) is reflexive, see also refrelressn 37933. (Contributed by Peter Mazsa, 12-Jun-2024.) |
⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥) | ||
Theorem | antisymressn 37853 | Every class ' R ' restricted to the singleton of the class ' A ' (see ressn2 37851) is antisymmetric. (Contributed by Peter Mazsa, 11-Jun-2024.) |
⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦) | ||
Theorem | trressn 37854 | Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 37851) is transitive, see also trrelressn 37992. (Contributed by Peter Mazsa, 16-Jun-2024.) |
⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧) | ||
Theorem | relbrcoss 37855* | 𝐴 and 𝐵 are cosets by relation 𝑅: a binary relation. (Contributed by Peter Mazsa, 22-Apr-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Rel 𝑅 → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅)))) | ||
Theorem | br1cossinres 37856* | 𝐵 and 𝐶 are cosets by an intersection with a restriction: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (𝑆 ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶)))) | ||
Theorem | br1cossxrnres 37857* | ⟨𝐵, 𝐶⟩ and ⟨𝐷, 𝐸⟩ are cosets by an range Cartesian product with a restriction: a binary relation. (Contributed by Peter Mazsa, 8-Jun-2021.) |
⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷)))) | ||
Theorem | br1cossinidres 37858* | 𝐵 and 𝐶 are cosets by an intersection with the restricted identity class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐶)))) | ||
Theorem | br1cossincnvepres 37859* | 𝐵 and 𝐶 are cosets by an intersection with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (◡ E ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶)))) | ||
Theorem | br1cossxrnidres 37860* | ⟨𝐵, 𝐶⟩ and ⟨𝐷, 𝐸⟩ are cosets by a range Cartesian product with the restricted identity class: a binary relation. (Contributed by Peter Mazsa, 8-Jun-2021.) |
⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( I ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷)))) | ||
Theorem | br1cossxrncnvepres 37861* | ⟨𝐵, 𝐶⟩ and ⟨𝐷, 𝐸⟩ are cosets by a range Cartesian product with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 12-May-2021.) |
⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ∈ 𝑢 ∧ 𝑢𝑅𝐷)))) | ||
Theorem | dmcoss3 37862 | The domain of cosets is the domain of converse. (Contributed by Peter Mazsa, 4-Jan-2019.) |
⊢ dom ≀ 𝑅 = dom ◡𝑅 | ||
Theorem | dmcoss2 37863 | The domain of cosets is the range. (Contributed by Peter Mazsa, 27-Dec-2018.) |
⊢ dom ≀ 𝑅 = ran 𝑅 | ||
Theorem | rncossdmcoss 37864 | The range of cosets is the domain of them (this should be rncoss 5969 but there exists a theorem with this name already). (Contributed by Peter Mazsa, 12-Dec-2019.) |
⊢ ran ≀ 𝑅 = dom ≀ 𝑅 | ||
Theorem | dm1cosscnvepres 37865 | The domain of cosets of the restricted converse epsilon relation is the union of the restriction. (Contributed by Peter Mazsa, 18-May-2019.) (Revised by Peter Mazsa, 26-Sep-2021.) |
⊢ dom ≀ (◡ E ↾ 𝐴) = ∪ 𝐴 | ||
Theorem | dmcoels 37866 | The domain of coelements in 𝐴 is the union of 𝐴. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Peter Mazsa, 5-Apr-2018.) (Revised by Peter Mazsa, 26-Sep-2021.) |
⊢ dom ∼ 𝐴 = ∪ 𝐴 | ||
Theorem | eldmcoss 37867* | Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) | ||
Theorem | eldmcoss2 37868 | Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 28-Dec-2018.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐴)) | ||
Theorem | eldm1cossres 37869* | Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.) |
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑢 ∈ 𝐴 𝑢𝑅𝐵)) | ||
Theorem | eldm1cossres2 37870* | Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.) |
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom ≀ (𝑅 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ [𝑥]𝑅)) | ||
Theorem | refrelcosslem 37871 | Lemma for the left side of the refrelcoss3 37872 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.) |
⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥 | ||
Theorem | refrelcoss3 37872* | The class of cosets by 𝑅 is reflexive, see dfrefrel3 37925. (Contributed by Peter Mazsa, 30-Jul-2019.) |
⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ∧ Rel ≀ 𝑅) | ||
Theorem | refrelcoss2 37873 | The class of cosets by 𝑅 is reflexive, see dfrefrel2 37924. (Contributed by Peter Mazsa, 30-Jul-2019.) |
⊢ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅) | ||
Theorem | symrelcoss3 37874 | The class of cosets by 𝑅 is symmetric, see dfsymrel3 37959. (Contributed by Peter Mazsa, 28-Mar-2019.) (Revised by Peter Mazsa, 17-Sep-2021.) |
⊢ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅) | ||
Theorem | symrelcoss2 37875 | The class of cosets by 𝑅 is symmetric, see dfsymrel2 37958. (Contributed by Peter Mazsa, 27-Dec-2018.) |
⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅) | ||
Theorem | cossssid 37876 | Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 27-Jul-2021.) |
⊢ ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅))) | ||
Theorem | cossssid2 37877* | Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 10-Mar-2019.) |
⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑥∀𝑦(∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) | ||
Theorem | cossssid3 37878* | Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 10-Mar-2019.) |
⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦)) | ||
Theorem | cossssid4 37879* | Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 31-Aug-2021.) |
⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑢∃*𝑥 𝑢𝑅𝑥) | ||
Theorem | cossssid5 37880* | Equivalent expressions for the class of cosets by 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ ( ≀ 𝑅 ⊆ I ↔ ∀𝑥 ∈ ran 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]◡𝑅 ∩ [𝑦]◡𝑅) = ∅)) | ||
Theorem | brcosscnv 37881* | 𝐴 and 𝐵 are cosets by converse 𝑅: a binary relation. (Contributed by Peter Mazsa, 23-Jan-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) | ||
Theorem | brcosscnv2 37882 | 𝐴 and 𝐵 are cosets by converse 𝑅: a binary relation. (Contributed by Peter Mazsa, 12-Mar-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑅𝐵 ↔ ([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅)) | ||
Theorem | br1cosscnvxrn 37883 | 𝐴 and 𝐵 are cosets by the converse range Cartesian product: a binary relation. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡(𝑅 ⋉ 𝑆)𝐵 ↔ (𝐴 ≀ ◡𝑅𝐵 ∧ 𝐴 ≀ ◡𝑆𝐵))) | ||
Theorem | 1cosscnvxrn 37884 | Cosets by the converse range Cartesian product. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.) |
⊢ ≀ ◡(𝐴 ⋉ 𝐵) = ( ≀ ◡𝐴 ∩ ≀ ◡𝐵) | ||
Theorem | cosscnvssid3 37885* | Equivalent expressions for the class of cosets by the converse of 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 28-Jul-2021.) |
⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣)) | ||
Theorem | cosscnvssid4 37886* | Equivalent expressions for the class of cosets by the converse of 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 31-Aug-2021.) |
⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥) | ||
Theorem | cosscnvssid5 37887* | Equivalent expressions for the class of cosets by the converse of the relation 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ (( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅) ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅)) | ||
Theorem | coss0 37888 | Cosets by the empty set are the empty set. (Contributed by Peter Mazsa, 22-Oct-2019.) |
⊢ ≀ ∅ = ∅ | ||
Theorem | cossid 37889 | Cosets by the identity relation are the identity relation. (Contributed by Peter Mazsa, 16-Jan-2019.) |
⊢ ≀ I = I | ||
Theorem | cosscnvid 37890 | Cosets by the converse identity relation are the identity relation. (Contributed by Peter Mazsa, 27-Sep-2021.) |
⊢ ≀ ◡ I = I | ||
Theorem | trcoss 37891* | Sufficient condition for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 26-Dec-2018.) |
⊢ (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧)) | ||
Theorem | eleccossin 37892 | Two ways of saying that the coset of 𝐴 and the coset of 𝐶 have the common element 𝐵. (Contributed by Peter Mazsa, 15-Oct-2021.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴 ≀ 𝑅𝐵 ∧ 𝐵 ≀ 𝑅𝐶))) | ||
Theorem | trcoss2 37893* | Equivalent expressions for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 4-Jul-2020.) (Revised by Peter Mazsa, 16-Oct-2021.) |
⊢ (∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ ∀𝑥∀𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅)) | ||
Definition | df-rels 37894 |
Define the relations class. Proper class relations (like I, see
reli 5822) are not elements of it. The element of this
class and the
relation predicate are the same when 𝑅 is a set (see elrelsrel 37896).
The class of relations is a great tool we can use when we define classes of different relations as nullary class constants as required by the 2. point in our Guidelines https://us.metamath.org/mpeuni/mathbox.html 37896. When we want to define a specific class of relations as a nullary class constant, the appropriate method is the following: 1. We define the specific nullary class constant for general sets (see e.g. df-refs 37919), then 2. we get the required class of relations by the intersection of the class of general sets above with the class of relations df-rels 37894 (see df-refrels 37920 and the resulting dfrefrels2 37922 and dfrefrels3 37923). 3. Finally, in order to be able to work with proper classes (like iprc 7913) as well, we define the predicate of the relation (see df-refrel 37921) so that it is true for the relevant proper classes (see refrelid 37931), and that the element of the class of the required relations (e.g. elrefrels3 37928) and this predicate are the same in case of sets (see elrefrelsrel 37929). (Contributed by Peter Mazsa, 13-Jun-2018.) |
⊢ Rels = 𝒫 (V × V) | ||
Theorem | elrels2 37895 | The element of the relations class (df-rels 37894) and the relation predicate (df-rel 5679) are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 14-Jun-2018.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V))) | ||
Theorem | elrelsrel 37896 | The element of the relations class (df-rels 37894) and the relation predicate are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 24-Nov-2018.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅)) | ||
Theorem | elrelsrelim 37897 | The element of the relations class is a relation. (Contributed by Peter Mazsa, 20-Jul-2019.) |
⊢ (𝑅 ∈ Rels → Rel 𝑅) | ||
Theorem | elrels5 37898 | Equivalent expressions for an element of the relations class. (Contributed by Peter Mazsa, 21-Jul-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ (𝑅 ↾ dom 𝑅) = 𝑅)) | ||
Theorem | elrels6 37899 | Equivalent expressions for an element of the relations class. (Contributed by Peter Mazsa, 21-Jul-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)) | ||
Theorem | elrelscnveq3 37900* | Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.) |
⊢ (𝑅 ∈ Rels → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) |
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