P. 377 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37601-37700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cosseqi 37601	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 37602	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 37603*	The class of cosets by a restriction. (Contributed by Peter Mazsa, 20-Apr-2019.)
⊢ ≀ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
Theorem	dfcoels 37604*	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 37605*	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 37606*	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 37607	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 37608	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 37609*	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 37610*	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 37611	The converse of cosets by 𝑅 are cosets by 𝑅. (Contributed by Peter Mazsa, 3-May-2019.)
⊢ ◡ ≀ 𝑅 = ≀ 𝑅
Theorem	br2coss 37612	Cosets by ≀ 𝑅 binary relation. (Contributed by Peter Mazsa, 25-Aug-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅))
Theorem	br1cossres 37613*	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 37614*	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 37615	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 37616*	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 37617*	Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 37616) is reflexive, see also refrelressn 37698. (Contributed by Peter Mazsa, 12-Jun-2024.)
⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)
Theorem	antisymressn 37618	Every class ' R ' restricted to the singleton of the class ' A ' (see ressn2 37616) is antisymmetric. (Contributed by Peter Mazsa, 11-Jun-2024.)
⊢ ∀𝑥∀𝑦((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑥) → 𝑥 = 𝑦)
Theorem	trressn 37619	Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 37616) is transitive, see also trrelressn 37757. (Contributed by Peter Mazsa, 16-Jun-2024.)
⊢ ∀𝑥∀𝑦∀𝑧((𝑥(𝑅 ↾ {𝐴})𝑦 ∧ 𝑦(𝑅 ↾ {𝐴})𝑧) → 𝑥(𝑅 ↾ {𝐴})𝑧)
Theorem	relbrcoss 37620*	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 37621*	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 37622*	⟨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 37623*	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 37624*	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 37625*	⟨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 37626*	⟨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 37627	The domain of cosets is the domain of converse. (Contributed by Peter Mazsa, 4-Jan-2019.)
⊢ dom ≀ 𝑅 = dom ◡𝑅
Theorem	dmcoss2 37628	The domain of cosets is the range. (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ dom ≀ 𝑅 = ran 𝑅
Theorem	rncossdmcoss 37629	The range of cosets is the domain of them (this should be rncoss 5971 but there exists a theorem with this name already). (Contributed by Peter Mazsa, 12-Dec-2019.)
⊢ ran ≀ 𝑅 = dom ≀ 𝑅
Theorem	dm1cosscnvepres 37630	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 37631	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 37632*	Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
Theorem	eldmcoss2 37633	Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 28-Dec-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐴))
Theorem	eldm1cossres 37634*	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 37635*	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 37636	Lemma for the left side of the refrelcoss3 37637 reflexivity theorem. (Contributed by Peter Mazsa, 1-Apr-2019.)
⊢ ∀𝑥 ∈ dom ≀ 𝑅𝑥 ≀ 𝑅𝑥
Theorem	refrelcoss3 37637*	The class of cosets by 𝑅 is reflexive, see dfrefrel3 37690. (Contributed by Peter Mazsa, 30-Jul-2019.)
⊢ (∀𝑥 ∈ dom ≀ 𝑅∀𝑦 ∈ ran ≀ 𝑅(𝑥 = 𝑦 → 𝑥 ≀ 𝑅𝑦) ∧ Rel ≀ 𝑅)
Theorem	refrelcoss2 37638	The class of cosets by 𝑅 is reflexive, see dfrefrel2 37689. (Contributed by Peter Mazsa, 30-Jul-2019.)
⊢ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
Theorem	symrelcoss3 37639	The class of cosets by 𝑅 is symmetric, see dfsymrel3 37724. (Contributed by Peter Mazsa, 28-Mar-2019.) (Revised by Peter Mazsa, 17-Sep-2021.)
⊢ (∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) ∧ Rel ≀ 𝑅)
Theorem	symrelcoss2 37640	The class of cosets by 𝑅 is symmetric, see dfsymrel2 37723. (Contributed by Peter Mazsa, 27-Dec-2018.)
⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
Theorem	cossssid 37641	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 37642*	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 37643*	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 37644*	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 37645*	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 37646*	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 37647	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 37648	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 37649	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 37650*	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 37651*	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 37652*	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 37653	Cosets by the empty set are the empty set. (Contributed by Peter Mazsa, 22-Oct-2019.)
⊢ ≀ ∅ = ∅
Theorem	cossid 37654	Cosets by the identity relation are the identity relation. (Contributed by Peter Mazsa, 16-Jan-2019.)
⊢ ≀ I = I
Theorem	cosscnvid 37655	Cosets by the converse identity relation are the identity relation. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ≀ ◡ I = I
Theorem	trcoss 37656*	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 37657	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 37658*	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) ≠ ∅))
21.24.5  Relations
Definition	df-rels 37659	Define the relations class. Proper class relations (like I, see reli 5826) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 37661). 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 37661. 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 37684), 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 37659 (see df-refrels 37685 and the resulting dfrefrels2 37687 and dfrefrels3 37688). 3. Finally, in order to be able to work with proper classes (like iprc 7908) as well, we define the predicate of the relation (see df-refrel 37686) so that it is true for the relevant proper classes (see refrelid 37696), and that the element of the class of the required relations (e.g. elrefrels3 37693) and this predicate are the same in case of sets (see elrefrelsrel 37694). (Contributed by Peter Mazsa, 13-Jun-2018.)
⊢ Rels = 𝒫 (V × V)
Theorem	elrels2 37660	The element of the relations class (df-rels 37659) and the relation predicate (df-rel 5683) are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 14-Jun-2018.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
Theorem	elrelsrel 37661	The element of the relations class (df-rels 37659) 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 37662	The element of the relations class is a relation. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ (𝑅 ∈ Rels → Rel 𝑅)
Theorem	elrels5 37663	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 37664	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 37665*	Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ (𝑅 ∈ Rels → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)))
Theorem	elrelscnveq 37666	Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅))
Theorem	elrelscnveq2 37667*	Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ (𝑅 ∈ Rels → (◡𝑅 = 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)))
Theorem	elrelscnveq4 37668*	Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)))
Theorem	cnvelrels 37669	The converse of a set is an element of the class of relations. (Contributed by Peter Mazsa, 18-Aug-2019.)
⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ Rels )
Theorem	cosselrels 37670	Cosets of sets are elements of the relations class. Implies ⊢ (𝑅 ∈ Rels → ≀ 𝑅 ∈ Rels ). (Contributed by Peter Mazsa, 25-Aug-2021.)
⊢ (𝐴 ∈ 𝑉 → ≀ 𝐴 ∈ Rels )
Theorem	cosscnvelrels 37671	Cosets of converse sets are elements of the relations class. (Contributed by Peter Mazsa, 31-Aug-2021.)
⊢ (𝐴 ∈ 𝑉 → ≀ ◡𝐴 ∈ Rels )
21.24.6  Subset relations
Definition	df-ssr 37672*	Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5580) and identity relation (df-id 5574) classes. Subset relation class and Scott Fenton's subset class df-sset 35133 are the same: S = SSet (compare dfssr2 37673 with df-sset 35133), the only reason we do not use dfssr2 37673 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 3965) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, see brssr 37675. Yet in general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for sets, see the comment of df-ss 3965. The only exception (aside from directly investigating the class S e.g. in relssr 37674 or in extssr 37683) is when we have a specific purpose with its usage, like in case of df-refs 37684 versus df-cnvrefs 37699, 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 37483, extep 37455 and extssr 37683, 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⟩ ∣ 𝑥 ⊆ 𝑦}
Theorem	dfssr2 37673	Alternate definition of the subset relation. (Contributed by Peter Mazsa, 9-Aug-2021.)
⊢ S = ((V × V) ∖ ran ( E ⋉ (V ∖ E )))
Theorem	relssr 37674	The subset relation is a relation. (Contributed by Peter Mazsa, 1-Aug-2019.)
⊢ Rel S
Theorem	brssr 37675	The subset relation and subclass relationship (df-ss 3965) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set. (Contributed by Peter Mazsa, 31-Jul-2019.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵))
Theorem	brssrid 37676	Any set is a subset of itself. (Contributed by Peter Mazsa, 1-Aug-2019.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 S 𝐴)
Theorem	issetssr 37677	Two ways of expressing set existence. (Contributed by Peter Mazsa, 1-Aug-2019.)
⊢ (𝐴 ∈ V ↔ 𝐴 S 𝐴)
Theorem	brssrres 37678	Restricted subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)
⊢ (𝐶 ∈ 𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶)))
Theorem	br1cnvssrres 37679	Restricted converse subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)
⊢ (𝐵 ∈ 𝑉 → (𝐵◡( S ↾ 𝐴)𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐵)))
Theorem	brcnvssr 37680	The converse of a subset relation swaps arguments. (Contributed by Peter Mazsa, 1-Aug-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐴◡ S 𝐵 ↔ 𝐵 ⊆ 𝐴))
Theorem	brcnvssrid 37681	Any set is a converse subset of itself. (Contributed by Peter Mazsa, 9-Jun-2021.)
⊢ (𝐴 ∈ 𝑉 → 𝐴◡ S 𝐴)
Theorem	br1cossxrncnvssrres 37682*	⟨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.)
⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (◡ S ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ⊆ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ⊆ 𝑢 ∧ 𝑢𝑅𝐷))))
Theorem	extssr 37683	Property of subset relation, see also extid 37483, extep 37455 and the comment of df-ssr 37672. (Contributed by Peter Mazsa, 10-Jul-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ S = [𝐵]◡ S ↔ 𝐴 = 𝐵))
21.24.7  Reflexivity
Definition	df-refs 37684	Define the class of all reflexive sets. It is used only by df-refrels 37685. We use subset relation S (df-ssr 37672) here to be able to define converse reflexivity (df-cnvrefs 37699), see also the comment of df-ssr 37672. The elements of this class are not necessarily relations (versus df-refrels 37685). Note the similarity of Definitions df-refs 37684, df-syms 37716 and df-trs 37746, cf. comments of dfrefrels2 37687. (Contributed by Peter Mazsa, 19-Jul-2019.)
⊢ Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
Definition	df-refrels 37685	Define the class of reflexive relations. This is practically dfrefrels2 37687 (which reveals that RefRels can not include proper classes like I as is elements, see comments of dfrefrels2 37687). Another alternative definition is dfrefrels3 37688. The element of this class and the reflexive relation predicate (df-refrel 37686) are the same, that is, (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝐴 is a set, see elrefrelsrel 37694. This definition is similar to the definitions of the classes of symmetric (df-symrels 37717) and transitive (df-trrels 37747) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ RefRels = ( Refs ∩ Rels )
Definition	df-refrel 37686	Define the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.) This is a surprising definition, see the comment of dfrefrel3 37690. Alternate definitions are dfrefrel2 37689 and dfrefrel3 37690. For sets, being an element of the class of reflexive relations (df-refrels 37685) is equivalent to satisfying the reflexive relation predicate, that is (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝑅 is a set, see elrefrelsrel 37694. (Contributed by Peter Mazsa, 16-Jul-2021.)
⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Theorem	dfrefrels2 37687	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 7908) 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 3492. 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 37686. See also the comment of df-rels 37659. Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 37687, it keeps restriction of I: this is why the very similar definitions df-refs 37684, df-syms 37716 and df-trs 37746 diverge when we switch from (general) sets to relations in dfrefrels2 37687, dfsymrels2 37719 and dftrrels2 37749. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
Theorem	dfrefrels3 37688*	Alternate definition of the class of reflexive relations. (Contributed by Peter Mazsa, 8-Jul-2019.)
⊢ RefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦)}
Theorem	dfrefrel2 37689	Alternate definition of the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
Theorem	dfrefrel3 37690*	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 7146 / idrefALT 6112 or df-reflexive 47901 ⊢ (𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴𝑥𝑅𝑥)). It turns out that the not-surprising definition which contains ∀𝑥 ∈ dom 𝑟𝑥𝑟𝑥 needs symmetry as well, see refsymrels3 37740. Only when this symmetry condition holds, like in case of equivalence relations, see dfeqvrels3 37763, 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 37493 where ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 = 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴𝑥𝑅𝑥). See also similar definition of the converse reflexive relations class dfcnvrefrel3 37705. (Contributed by Peter Mazsa, 8-Jul-2019.)
⊢ ( RefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) ∧ Rel 𝑅))
Theorem	dfrefrel5 37691*	Alternate definition of the reflexive relation predicate. (Contributed by Peter Mazsa, 12-Dec-2023.)
⊢ ( RefRel 𝑅 ↔ (∀𝑥 ∈ (dom 𝑅 ∩ ran 𝑅)𝑥𝑅𝑥 ∧ Rel 𝑅))
Theorem	elrefrels2 37692	Element of the class of reflexive relations. (Contributed by Peter Mazsa, 23-Jul-2019.)
⊢ (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))
Theorem	elrefrels3 37693*	Element of the class of reflexive relations. (Contributed by Peter Mazsa, 23-Jul-2019.)
⊢ (𝑅 ∈ RefRels ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) ∧ 𝑅 ∈ Rels ))
Theorem	elrefrelsrel 37694	For sets, being an element of the class of reflexive relations (df-refrels 37685) is equivalent to satisfying the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))
Theorem	refreleq 37695	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))
Theorem	refrelid 37696	Identity relation is reflexive. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ RefRel I
Theorem	refrelcoss 37697	The class of cosets by 𝑅 is reflexive. (Contributed by Peter Mazsa, 4-Jul-2020.)
⊢ RefRel ≀ 𝑅
Theorem	refrelressn 37698	Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 37616) is reflexive. (Contributed by Peter Mazsa, 12-Jun-2024.)
⊢ (𝐴 ∈ 𝑉 → RefRel (𝑅 ↾ {𝐴}))
21.24.8  Converse reflexivity
Definition	df-cnvrefs 37699	Define the class of all converse reflexive sets, see the comment of df-ssr 37672. It is used only by df-cnvrefrels 37700. (Contributed by Peter Mazsa, 22-Jul-2019.)
⊢ CnvRefs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥))◡ S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
Definition	df-cnvrefrels 37700	Define the class of converse reflexive relations. This is practically dfcnvrefrels2 37702 (which uses the traditional subclass relation ⊆) : we use converse subset relation (brcnvssr 37680) here to ensure the comparability to the definitions of the classes of all reflexive (df-ref 23230), symmetric (df-syms 37716) and transitive (df-trs 37746) sets. We use this concept to define functions (df-funsALTV 37855, df-funALTV 37856) and disjoints (df-disjs 37878, df-disjALTV 37879). For sets, being an element of the class of converse reflexive relations is equivalent to satisfying the converse reflexive relation predicate, see elcnvrefrelsrel 37710. Alternate definitions are dfcnvrefrels2 37702 and dfcnvrefrels3 37703. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ CnvRefRels = ( CnvRefs ∩ Rels )