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Theorem List for Metamath Proof Explorer - 35701-35800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcosscnvex 35701 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.)
(𝐴𝑉 → ≀ 𝐴 ∈ V)

Theorem1cosscnvepresex 35702 Sufficient condition for a restricted converse epsilon coset to be a set. (Contributed by Peter Mazsa, 24-Sep-2021.)
(𝐴𝑉 → ≀ ( E ↾ 𝐴) ∈ V)

Theorem1cossxrncnvepresex 35703 Sufficient condition for a restricted converse epsilon range Cartesian product to be a set. (Contributed by Peter Mazsa, 23-Sep-2021.)
((𝐴𝑉𝑅𝑊) → ≀ (𝑅 ⋉ ( E ↾ 𝐴)) ∈ V)

Theoremrelcoss 35704 Cosets by 𝑅 is a relation. (Contributed by Peter Mazsa, 27-Dec-2018.)
Rel ≀ 𝑅

Theoremrelcoels 35705 Coelements on 𝐴 is a relation. (Contributed by Peter Mazsa, 5-Oct-2021.)
Rel ∼ 𝐴

Theoremcossss 35706 Subclass theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 11-Nov-2019.)
(𝐴𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)

Theoremcosseq 35707 Equality theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 9-Jan-2018.)
(𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵)

Theoremcosseqi 35708 Equality theorem for the classes of cosets by 𝐴 and 𝐵, inference form. (Contributed by Peter Mazsa, 9-Jan-2018.)
𝐴 = 𝐵       𝐴 = ≀ 𝐵

Theoremcosseqd 35709 Equality theorem for the classes of cosets by 𝐴 and 𝐵, deduction form. (Contributed by Peter Mazsa, 4-Nov-2019.)
(𝜑𝐴 = 𝐵)       (𝜑 → ≀ 𝐴 = ≀ 𝐵)

Theorem1cossres 35710* The class of cosets by a restriction. (Contributed by Peter Mazsa, 20-Apr-2019.)
≀ (𝑅𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑢𝑅𝑥𝑢𝑅𝑦)}

Theoremdfcoels 35711* Alternate definition of the class of coelements on the class 𝐴. (Contributed by Peter Mazsa, 20-Apr-2019.)
𝐴 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝐴 (𝑥𝑢𝑦𝑢)}

Theorembrcoss 35712* 𝐴 and 𝐵 are cosets by 𝑅: a binary relation. (Contributed by Peter Mazsa, 27-Dec-2018.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))

Theorembrcoss2 35713* Alternate form of the 𝐴 and 𝐵 are cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 26-Mar-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝐴 ∈ [𝑢]𝑅𝐵 ∈ [𝑢]𝑅)))

Theorembrcoss3 35714 Alternate form of the 𝐴 and 𝐵 are cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 26-Mar-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅))

Theorembrcosscnvcoss 35715 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.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵𝐵𝑅𝐴))

Theorembrcoels 35716* 𝐵 and 𝐶 are coelements : a binary relation. (Contributed by Peter Mazsa, 14-Jan-2020.) (Revised by Peter Mazsa, 5-Oct-2021.)
((𝐵𝑉𝐶𝑊) → (𝐵𝐴𝐶 ↔ ∃𝑢𝐴 (𝐵𝑢𝐶𝑢)))

Theoremcocossss 35717* Two ways of saying that cosets by cosets by 𝑅 is a subclass. (Contributed by Peter Mazsa, 17-Sep-2021.)
( ≀ ≀ 𝑅𝑆 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))

Theoremcnvcosseq 35718 The converse of cosets by 𝑅 are cosets by 𝑅. (Contributed by Peter Mazsa, 3-May-2019.)
𝑅 = ≀ 𝑅

Theorembr2coss 35719 Cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 25-Aug-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅))

Theorembr1cossres 35720* 𝐵 and 𝐶 are cosets by a restriction: a binary relation. (Contributed by Peter Mazsa, 30-Dec-2018.)
((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅𝐴)𝐶 ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑅𝐶)))

Theorembr1cossres2 35721* 𝐵 and 𝐶 are cosets by a restriction: a binary relation. (Contributed by Peter Mazsa, 3-Jan-2018.)
((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅𝐴)𝐶 ↔ ∃𝑥𝐴 (𝐵 ∈ [𝑥]𝑅𝐶 ∈ [𝑥]𝑅)))

Theoremrelbrcoss 35722* 𝐴 and 𝐵 are cosets by relation 𝑅: a binary relation. (Contributed by Peter Mazsa, 22-Apr-2021.)
((𝐴𝑉𝐵𝑊) → (Rel 𝑅 → (𝐴𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅))))

Theorembr1cossinres 35723* 𝐵 and 𝐶 are cosets by an intersection with a restriction: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ (𝑆𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝑢𝑆𝐵𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶𝑢𝑅𝐶))))

Theorembr1cossxrnres 35724* 𝐵, 𝐶 and 𝐷, 𝐸 are cosets by an intersection with a restriction: a binary relation. (Contributed by Peter Mazsa, 8-Jun-2021.)
(((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢𝑆𝐶𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸𝑢𝑅𝐷))))

Theorembr1cossinidres 35725* 𝐵 and 𝐶 are cosets by an intersection with the restricted identity class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝑢 = 𝐵𝑢𝑅𝐵) ∧ (𝑢 = 𝐶𝑢𝑅𝐶))))

Theorembr1cossincnvepres 35726* 𝐵 and 𝐶 are cosets by an intersection with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ ( E ↾ 𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝐵𝑢𝑢𝑅𝐵) ∧ (𝐶𝑢𝑢𝑅𝐶))))

Theorembr1cossxrnidres 35727* 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.)
(((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( I ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝑢 = 𝐶𝑢𝑅𝐵) ∧ (𝑢 = 𝐸𝑢𝑅𝐷))))

Theorembr1cossxrncnvepres 35728* 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.)
(((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( E ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝐶𝑢𝑢𝑅𝐵) ∧ (𝐸𝑢𝑢𝑅𝐷))))

Theoremdmcoss3 35729 The domain of cosets is the domain of converse. (Contributed by Peter Mazsa, 4-Jan-2019.)
dom ≀ 𝑅 = dom 𝑅

Theoremdmcoss2 35730 The domain of cosets is the range. (Contributed by Peter Mazsa, 27-Dec-2018.)
dom ≀ 𝑅 = ran 𝑅

Theoremrncossdmcoss 35731 The range of cosets is the domain of them (this should be rncoss 5819 but there exists a theorem with this name already). (Contributed by Peter Mazsa, 12-Dec-2019.)
ran ≀ 𝑅 = dom ≀ 𝑅

Theoremdm1cosscnvepres 35732 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

Theoremdmcoels 35733 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

Theoremeldmcoss 35734* Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.)
(𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))

Theoremeldmcoss2 35735 Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 28-Dec-2018.)
(𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅𝐴𝑅𝐴))

Theoremeldm1cossres 35736* Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.)
(𝐵𝑉 → (𝐵 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑢𝐴 𝑢𝑅𝐵))

Theoremeldm1cossres2 35737* Elementhood in the domain of restricted cosets. (Contributed by Peter Mazsa, 30-Dec-2018.)
(𝐵𝑉 → (𝐵 ∈ dom ≀ (𝑅𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ [𝑥]𝑅))

Theoremrefrelcosslem 35738 Lemma for the left side of the refrelcoss3 35739 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.)
𝑥 ∈ dom ≀ 𝑅𝑥𝑅𝑥

Theoremrefrelcoss3 35739* The class of cosets by 𝑅 is reflexive, see dfrefrel3 35792. (Contributed by Peter Mazsa, 30-Jul-2019.)
(∀𝑥 ∈ dom ≀ 𝑅𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel ≀ 𝑅)

Theoremrefrelcoss2 35740 The class of cosets by 𝑅 is reflexive, see dfrefrel2 35791. (Contributed by Peter Mazsa, 30-Jul-2019.)
(( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)

Theoremsymrelcoss3 35741 The class of cosets by 𝑅 is symmetric, see dfsymrel3 35822. (Contributed by Peter Mazsa, 28-Mar-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)
(∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel ≀ 𝑅)

Theoremsymrelcoss2 35742 The class of cosets by 𝑅 is symmetric, see dfsymrel2 35821. (Contributed by Peter Mazsa, 27-Dec-2018.)
(𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)

Theoremcossssid 35743 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)))

Theoremcossssid2 35744* 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 ↔ ∀𝑥𝑦(∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))

Theoremcossssid3 35745* 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 ↔ ∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))

Theoremcossssid4 35746* 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)

Theoremcossssid5 35747* 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 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅))

Theorembrcosscnv 35748* 𝐴 and 𝐵 are cosets by converse 𝑅: a binary relation. (Contributed by Peter Mazsa, 23-Jan-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))

Theorembrcosscnv2 35749 𝐴 and 𝐵 are cosets by converse 𝑅: a binary relation. (Contributed by Peter Mazsa, 12-Mar-2019.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅))

Theorembr1cosscnvxrn 35750 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.)
((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵)))

Theorem1cosscnvxrn 35751 Cosets by the converse range Cartesian product. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
(𝐴𝐵) = ( ≀ 𝐴 ∩ ≀ 𝐵)

Theoremcosscnvssid3 35752* 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 ↔ ∀𝑢𝑣𝑥((𝑢𝑅𝑥𝑣𝑅𝑥) → 𝑢 = 𝑣))

Theoremcosscnvssid4 35753* 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)

Theoremcosscnvssid5 35754* 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 𝑅𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅))

Theoremcoss0 35755 Cosets by the empty set are the empty set. (Contributed by Peter Mazsa, 22-Oct-2019.)
≀ ∅ = ∅

Theoremcossid 35756 Cosets by the identity relation are the identity relation. (Contributed by Peter Mazsa, 16-Jan-2019.)
≀ I = I

Theoremcosscnvid 35757 Cosets by the converse identity relation are the identity relation. (Contributed by Peter Mazsa, 27-Sep-2021.)
I = I

Theoremtrcoss 35758* Sufficient condition for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 26-Dec-2018.)
(∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Theoremeleccossin 35759 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.)
((𝐵𝑉𝐶𝑊) → (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))

Theoremtrcoss2 35760* Equivalent expressions for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 4-Jul-2020.) (Revised by Peter Mazsa, 16-Oct-2021.)
(∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))

20.22.5  Relations

Definitiondf-rels 35761 Define the relations class. Proper class relations (like I, see reli 5674) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 35763).

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 35763. 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 35786), 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 35761 (see df-refrels 35787 and the resulting dfrefrels2 35789 and dfrefrels3 35790).

3. Finally, in order to be able to work with proper classes (like iprc 7596) as well, we define the predicate of the relation (see df-refrel 35788) so that it is true for the relevant proper classes (see refrelid 35797), and that the element of the class of the required relations (e.g. elrefrels3 35794) and this predicate are the same in case of sets (see elrefrelsrel 35795). (Contributed by Peter Mazsa, 13-Jun-2018.)

Rels = 𝒫 (V × V)

Theoremelrels2 35762 The element of the relations class (df-rels 35761) and the relation predicate (df-rel 5538) are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 14-Jun-2018.)
(𝑅𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))

Theoremelrelsrel 35763 The element of the relations class (df-rels 35761) and the relation predicate are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 24-Nov-2018.)
(𝑅𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))

Theoremelrelsrelim 35764 The element of the relations class is a relation. (Contributed by Peter Mazsa, 20-Jul-2019.)
(𝑅 ∈ Rels → Rel 𝑅)

Theoremelrels5 35765 Equivalent expressions for an element of the relations class. (Contributed by Peter Mazsa, 21-Jul-2021.)
(𝑅𝑉 → (𝑅 ∈ Rels ↔ (𝑅 ↾ dom 𝑅) = 𝑅))

Theoremelrels6 35766 Equivalent expressions for an element of the relations class. (Contributed by Peter Mazsa, 21-Jul-2021.)
(𝑅𝑉 → (𝑅 ∈ Rels ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅))

Theoremelrelscnveq3 35767* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))

Theoremelrelscnveq 35768 Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ Rels → (𝑅𝑅𝑅 = 𝑅))

Theoremelrelscnveq2 35769* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))

Theoremelrelscnveq4 35770* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))

Theoremcnvelrels 35771 The converse of a set is an element of the class of relations. (Contributed by Peter Mazsa, 18-Aug-2019.)
(𝐴𝑉𝐴 ∈ Rels )

Theoremcosselrels 35772 Cosets of sets are elements of the relations class. Implies (𝑅 ∈ Rels → ≀ 𝑅 ∈ Rels ). (Contributed by Peter Mazsa, 25-Aug-2021.)
(𝐴𝑉 → ≀ 𝐴 ∈ Rels )

Theoremcosscnvelrels 35773 Cosets of converse sets are elements of the relations class. (Contributed by Peter Mazsa, 31-Aug-2021.)
(𝐴𝑉 → ≀ 𝐴 ∈ Rels )

20.22.6  Subset relations

Definitiondf-ssr 35774* Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5441) and identity relation (df-id 5436) classes. Subset relation class and Scott Fenton's subset class df-sset 33325 are the same: S = SSet (compare dfssr2 35775 with df-sset 33325), the only reason we do not use dfssr2 35775 as the base definition of the subsets class is the way we defined the epsilon relation and the identity relation classes.

The binary relation on the class of subsets and the subclass relationship (df-ss 3930) are the same, that is, (𝐴 S 𝐵𝐴𝐵) when 𝐵 is a set, see brssr 35777. Yet in general we use the subclass relation 𝐴𝐵 both for classes and for sets, see the comment of df-ss 3930. The only exception (aside from directly investigating the class S e.g. in relssr 35776 or in extssr 35785) is when we have a specific purpose with its usage, like in case of df-refs 35786 versus df-cnvrefs 35799, where we need S to define the class of reflexive sets in order to be able to define the class of converse reflexive sets with the help of the converse of S.

The subsets class S has another place in set.mm as well: if we define extensional relation based on the common property in extid 35604, extep 35576 and extssr 35785, then "extrelssr" " |- ExtRel S " is a theorem along with "extrelep" " |- ExtRel E " and "extrelid" " |- ExtRel I " . (Contributed by Peter Mazsa, 25-Jul-2019.)

S = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}

Theoremdfssr2 35775 Alternate definition of the subset relation. (Contributed by Peter Mazsa, 9-Aug-2021.)
S = ((V × V) ∖ ran ( E ⋉ (V ∖ E )))

Theoremrelssr 35776 The subset relation is a relation. (Contributed by Peter Mazsa, 1-Aug-2019.)
Rel S

Theorembrssr 35777 The subset relation and subclass relationship (df-ss 3930) are the same, that is, (𝐴 S 𝐵𝐴𝐵) when 𝐵 is a set. (Contributed by Peter Mazsa, 31-Jul-2019.)
(𝐵𝑉 → (𝐴 S 𝐵𝐴𝐵))

Theorembrssrid 35778 Any set is a subset of itself. (Contributed by Peter Mazsa, 1-Aug-2019.)
(𝐴𝑉𝐴 S 𝐴)

Theoremissetssr 35779 Two ways of expressing set existence. (Contributed by Peter Mazsa, 1-Aug-2019.)
(𝐴 ∈ V ↔ 𝐴 S 𝐴)

Theorembrssrres 35780 Restricted subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)
(𝐶𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐵𝐶)))

Theorembr1cnvssrres 35781 Restricted converse subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)
(𝐵𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐶𝐴𝐶𝐵)))

Theorembrcnvssr 35782 The converse of a subset relation swaps arguments. (Contributed by Peter Mazsa, 1-Aug-2019.)
(𝐴𝑉 → (𝐴 S 𝐵𝐵𝐴))

Theorembrcnvssrid 35783 Any set is a converse subset of itself. (Contributed by Peter Mazsa, 9-Jun-2021.)
(𝐴𝑉𝐴 S 𝐴)

Theorembr1cossxrncnvssrres 35784* 𝐵, 𝐶 and 𝐷, 𝐸 are cosets by range Cartesian product with restricted converse subsets class: a binary relation. (Contributed by Peter Mazsa, 9-Jun-2021.)
(((𝐵𝑉𝐶𝑊) ∧ (𝐷𝑋𝐸𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ ( S ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢𝐴 ((𝐶𝑢𝑢𝑅𝐵) ∧ (𝐸𝑢𝑢𝑅𝐷))))

Theoremextssr 35785 Property of subset relation, see also extid 35604, extep 35576 and the comment of df-ssr 35774. (Contributed by Peter Mazsa, 10-Jul-2019.)
((𝐴𝑉𝐵𝑊) → ([𝐴] S = [𝐵] S ↔ 𝐴 = 𝐵))

20.22.7  Reflexivity

Definitiondf-refs 35786 Define the class of all reflexive sets. It is used only by df-refrels 35787. We use subset relation S (df-ssr 35774) here to be able to define converse reflexivity (df-cnvrefs 35799), see also the comment of df-ssr 35774. The elements of this class are not necessarily relations (versus df-refrels 35787).

Note the similarity of the definitions df-refs 35786, df-syms 35814 and df-trs 35844, cf. comments of dfrefrels2 35789. (Contributed by Peter Mazsa, 19-Jul-2019.)

Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}

Definitiondf-refrels 35787 Define the class of reflexive relations. This is practically dfrefrels2 35789 (which reveals that RefRels can not include proper classes like I as is elements, see comments of dfrefrels2 35789).

Another alternative definition is dfrefrels3 35790. The element of this class and the reflexive relation predicate (df-refrel 35788) are the same, that is, (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝐴 is a set, see elrefrelsrel 35795.

This definition is similar to the definitions of the classes of symmetric (df-symrels 35815) and transitive (df-trrels 35845) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)

RefRels = ( Refs ∩ Rels )

Definitiondf-refrel 35788 Define the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.) This is a surprising definition, see the comment of dfrefrel3 35792. Alternate definitions are dfrefrel2 35791 and dfrefrel3 35792. For sets, being an element of the class of reflexive relations (df-refrels 35787) is equivalent to satisfying the reflexive relation predicate, that is (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝑅 is a set, see elrefrelsrel 35795. (Contributed by Peter Mazsa, 16-Jul-2021.)
( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))

Theoremdfrefrels2 35789 Alternate definition of the class of reflexive relations. This is a 0-ary class constant, which is recommended for definitions (see the 1. Guideline at https://us.metamath.org/ileuni/mathbox.html). Proper classes (like I, see iprc 7596) are not elements of this (or any) class: if a class is an element of another class, it is not a proper class but a set, see elex 3491. So if we use 0-ary constant classes as our main definitions, they are valid only for sets, not for proper classes. For proper classes we use predicate-type definitions like df-refrel 35788. See also the comment of df-rels 35761.

Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 35789, it keeps restriction of I: this is why the very similar definitions df-refs 35786, df-syms 35814 and df-trs 35844 diverge when we switch from (general) sets to relations in dfrefrels2 35789, dfsymrels2 35817 and dftrrels2 35847. (Contributed by Peter Mazsa, 20-Jul-2019.)

RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}

Theoremdfrefrels3 35790* Alternate definition of the class of reflexive relations. (Contributed by Peter Mazsa, 8-Jul-2019.)
RefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦)}

Theoremdfrefrel2 35791 Alternate definition of the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))

Theoremdfrefrel3 35792* Alternate definition of the reflexive relation predicate. A relation is reflexive iff: for all elements on its domain and range, if an element of its domain is the same as an element of its range, then there is the relation between them.

Note that this is definitely not the definition we are accustomed to, like e.g. idref 6884 / idrefALT 5949 or df-reflexive 45054 (𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥𝐴𝑥𝑅𝑥)). It turns out that the not-surprising definition which contains 𝑥 ∈ dom 𝑟𝑥𝑟𝑥 needs symmetry as well, see refsymrels3 35838. Only when this symmetry condition holds, like in case of equivalence relations, see dfeqvrels3 35860, can we write the traditional form 𝑥 ∈ dom 𝑟𝑥𝑟𝑥 for reflexive relations. For the special case with square Cartesian product when the two forms are equivalent see idinxpssinxp4 35613 where (∀𝑥𝐴𝑦𝐴(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴𝑥𝑅𝑥). See also similar definition of the converse reflexive relations class dfcnvrefrel3 35805. (Contributed by Peter Mazsa, 8-Jul-2019.)

( RefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel 𝑅))

Theoremelrefrels2 35793 Element of the class of reflexive relations. (Contributed by Peter Mazsa, 23-Jul-2019.)
(𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅 ∈ Rels ))

Theoremelrefrels3 35794* Element of the class of reflexive relations. (Contributed by Peter Mazsa, 23-Jul-2019.)
(𝑅 ∈ RefRels ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ 𝑅 ∈ Rels ))

Theoremelrefrelsrel 35795 For sets, being an element of the class of reflexive relations (df-refrels 35787) is equivalent to satisfying the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
(𝑅𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))

Theoremrefreleq 35796 Equality theorem for reflexive relation. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.)
(𝑅 = 𝑆 → ( RefRel 𝑅 ↔ RefRel 𝑆))

Theoremrefrelid 35797 Identity relation is reflexive. (Contributed by Peter Mazsa, 25-Jul-2021.)
RefRel I

Theoremrefrelcoss 35798 The class of cosets by 𝑅 is reflexive. (Contributed by Peter Mazsa, 4-Jul-2020.)
RefRel ≀ 𝑅

20.22.8  Converse reflexivity

Definitiondf-cnvrefs 35799 Define the class of all converse reflexive sets, see the comment of df-ssr 35774. It is used only by df-cnvrefrels 35800. (Contributed by Peter Mazsa, 22-Jul-2019.)
CnvRefs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}

Definitiondf-cnvrefrels 35800 Define the class of converse reflexive relations. This is practically dfcnvrefrels2 35802 (which uses the traditional subclass relation ) : we use converse subset relation (brcnvssr 35782) here to ensure the comparability to the definitions of the classes of all reflexive (df-ref 22089), symmetric (df-syms 35814) and transitive (df-trs 35844) sets.

We use this concept to define functions (df-funsALTV 35950, df-funALTV 35951) and disjoints (df-disjs 35973, df-disjALTV 35974).

For sets, being an element of the class of converse reflexive relations is equivalent to satisfying the converse reflexive relation predicate, see elcnvrefrelsrel 35808. Alternate definitions are dfcnvrefrels2 35802 and dfcnvrefrels3 35803. (Contributed by Peter Mazsa, 7-Jul-2019.)

CnvRefRels = ( CnvRefs ∩ Rels )

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