P. 386 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38501-38600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	relssr 38501	The subset relation is a relation. (Contributed by Peter Mazsa, 1-Aug-2019.)
⊢ Rel S
Theorem	brssr 38502	The subset relation and subclass relationship (df-ss 3968) 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 38503	Any set is a subset of itself. (Contributed by Peter Mazsa, 1-Aug-2019.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 S 𝐴)
Theorem	issetssr 38504	Two ways of expressing set existence. (Contributed by Peter Mazsa, 1-Aug-2019.)
⊢ (𝐴 ∈ V ↔ 𝐴 S 𝐴)
Theorem	brssrres 38505	Restricted subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)
⊢ (𝐶 ∈ 𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶)))
Theorem	br1cnvssrres 38506	Restricted converse subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)
⊢ (𝐵 ∈ 𝑉 → (𝐵◡( S ↾ 𝐴)𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐵)))
Theorem	brcnvssr 38507	The converse of a subset relation swaps arguments. (Contributed by Peter Mazsa, 1-Aug-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐴◡ S 𝐵 ↔ 𝐵 ⊆ 𝐴))
Theorem	brcnvssrid 38508	Any set is a converse subset of itself. (Contributed by Peter Mazsa, 9-Jun-2021.)
⊢ (𝐴 ∈ 𝑉 → 𝐴◡ S 𝐴)
Theorem	br1cossxrncnvssrres 38509*	⟨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 38510	Property of subset relation, see also extid 38311, extep 38284 and the comment of df-ssr 38499. (Contributed by Peter Mazsa, 10-Jul-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ S = [𝐵]◡ S ↔ 𝐴 = 𝐵))
21.26.7  Reflexivity
Definition	df-refs 38511	Define the class of all reflexive sets. It is used only by df-refrels 38512. We use subset relation S (df-ssr 38499) here to be able to define converse reflexivity (df-cnvrefs 38526), see also the comment of df-ssr 38499. The elements of this class are not necessarily relations (versus df-refrels 38512). Note the similarity of Definitions df-refs 38511, df-syms 38543 and df-trs 38573, cf. comments of dfrefrels2 38514. (Contributed by Peter Mazsa, 19-Jul-2019.)
⊢ Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
Definition	df-refrels 38512	Define the class of reflexive relations. This is practically dfrefrels2 38514 (which reveals that RefRels can not include proper classes like I as is elements, see comments of dfrefrels2 38514). Another alternative definition is dfrefrels3 38515. The element of this class and the reflexive relation predicate (df-refrel 38513) are the same, that is, (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝐴 is a set, see elrefrelsrel 38521. This definition is similar to the definitions of the classes of symmetric (df-symrels 38544) and transitive (df-trrels 38574) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ RefRels = ( Refs ∩ Rels )
Definition	df-refrel 38513	Define the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.) This is a surprising definition, see the comment of dfrefrel3 38517. Alternate definitions are dfrefrel2 38516 and dfrefrel3 38517. For sets, being an element of the class of reflexive relations (df-refrels 38512) is equivalent to satisfying the reflexive relation predicate, that is (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝑅 is a set, see elrefrelsrel 38521. (Contributed by Peter Mazsa, 16-Jul-2021.)
⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Theorem	dfrefrels2 38514	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 7933) 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 3501. 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 38513. See also the comment of df-rels 38486. Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 38514, it keeps restriction of I: this is why the very similar definitions df-refs 38511, df-syms 38543 and df-trs 38573 diverge when we switch from (general) sets to relations in dfrefrels2 38514, dfsymrels2 38546 and dftrrels2 38576. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
Theorem	dfrefrels3 38515*	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 38516	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 38517*	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 7166 / idrefALT 6131 or df-reflexive 49287 ⊢ (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 38567. Only when this symmetry condition holds, like in case of equivalence relations, see dfeqvrels3 38590, 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 38321 where ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 = 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴𝑥𝑅𝑥). See also similar definition of the converse reflexive relations class dfcnvrefrel3 38532. (Contributed by Peter Mazsa, 8-Jul-2019.)
⊢ ( RefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) ∧ Rel 𝑅))
Theorem	dfrefrel5 38518*	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 38519	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 38520*	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 38521	For sets, being an element of the class of reflexive relations (df-refrels 38512) is equivalent to satisfying the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))
Theorem	refreleq 38522	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 38523	Identity relation is reflexive. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ RefRel I
Theorem	refrelcoss 38524	The class of cosets by 𝑅 is reflexive. (Contributed by Peter Mazsa, 4-Jul-2020.)
⊢ RefRel ≀ 𝑅
Theorem	refrelressn 38525	Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 38443) is reflexive. (Contributed by Peter Mazsa, 12-Jun-2024.)
⊢ (𝐴 ∈ 𝑉 → RefRel (𝑅 ↾ {𝐴}))
21.26.8  Converse reflexivity
Definition	df-cnvrefs 38526	Define the class of all converse reflexive sets, see the comment of df-ssr 38499. It is used only by df-cnvrefrels 38527. (Contributed by Peter Mazsa, 22-Jul-2019.)
⊢ CnvRefs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥))◡ S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
Definition	df-cnvrefrels 38527	Define the class of converse reflexive relations. This is practically dfcnvrefrels2 38529 (which uses the traditional subclass relation ⊆) : we use converse subset relation (brcnvssr 38507) here to ensure the comparability to the definitions of the classes of all reflexive (df-ref 23513), symmetric (df-syms 38543) and transitive (df-trs 38573) sets. We use this concept to define functions (df-funsALTV 38682, df-funALTV 38683) and disjoints (df-disjs 38705, df-disjALTV 38706). For sets, being an element of the class of converse reflexive relations is equivalent to satisfying the converse reflexive relation predicate, see elcnvrefrelsrel 38537. Alternate definitions are dfcnvrefrels2 38529 and dfcnvrefrels3 38530. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ CnvRefRels = ( CnvRefs ∩ Rels )
Definition	df-cnvrefrel 38528	Define the converse reflexive relation predicate (read: 𝑅 is a converse reflexive relation), see also the comment of dfcnvrefrel3 38532. Alternate definitions are dfcnvrefrel2 38531 and dfcnvrefrel3 38532. (Contributed by Peter Mazsa, 16-Jul-2021.)
⊢ ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Theorem	dfcnvrefrels2 38529	Alternate definition of the class of converse reflexive relations. See the comment of dfrefrels2 38514. (Contributed by Peter Mazsa, 21-Jul-2021.)
⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}
Theorem	dfcnvrefrels3 38530*	Alternate definition of the class of converse reflexive relations. (Contributed by Peter Mazsa, 22-Jul-2019.)
⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)}
Theorem	dfcnvrefrel2 38531	Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 24-Jul-2019.)
⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Theorem	dfcnvrefrel3 38532*	Alternate definition of the converse reflexive relation predicate. A relation is converse reflexive iff: for all elements on its domain and range, if for an element of its domain and for an element of its range there is the relation between them, then the two elements are the same, cf. the comment of dfrefrel3 38517. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ ( CnvRefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ Rel 𝑅))
Theorem	dfcnvrefrel4 38533	Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-May-2024.)
⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ I ∧ Rel 𝑅))
Theorem	dfcnvrefrel5 38534*	Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-May-2024.)
⊢ ( CnvRefRel 𝑅 ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ Rel 𝑅))
Theorem	elcnvrefrels2 38535	Element of the class of converse reflexive relations. (Contributed by Peter Mazsa, 25-Jul-2019.)
⊢ (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels ))
Theorem	elcnvrefrels3 38536*	Element of the class of converse reflexive relations. (Contributed by Peter Mazsa, 30-Aug-2021.)
⊢ (𝑅 ∈ CnvRefRels ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ 𝑅 ∈ Rels ))
Theorem	elcnvrefrelsrel 38537	For sets, being an element of the class of converse reflexive relations (df-cnvrefrels 38527) is equivalent to satisfying the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅))
Theorem	cnvrefrelcoss2 38538	Necessary and sufficient condition for a coset relation to be a converse reflexive relation. (Contributed by Peter Mazsa, 27-Jul-2021.)
⊢ ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ I )
Theorem	cosselcnvrefrels2 38539	Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 25-Aug-2021.)
⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels ))
Theorem	cosselcnvrefrels3 38540*	Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 30-Aug-2021.)
⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝑅 ∈ Rels ))
Theorem	cosselcnvrefrels4 38541*	Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 31-Aug-2021.)
⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ (∀𝑢∃*𝑥 𝑢𝑅𝑥 ∧ ≀ 𝑅 ∈ Rels ))
Theorem	cosselcnvrefrels5 38542*	Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 5-Sep-2021.)
⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ (∀𝑥 ∈ ran 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]◡𝑅 ∩ [𝑦]◡𝑅) = ∅) ∧ ≀ 𝑅 ∈ Rels ))
21.26.9  Symmetry
Definition	df-syms 38543	Define the class of all symmetric sets. It is used only by df-symrels 38544. Note the similarity of Definitions df-refs 38511, df-syms 38543 and df-trs 38573, cf. the comment of dfrefrels2 38514. (Contributed by Peter Mazsa, 19-Jul-2019.)
⊢ Syms = {𝑥 ∣ ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
Definition	df-symrels 38544	Define the class of symmetric relations. For sets, being an element of the class of symmetric relations is equivalent to satisfying the symmetric relation predicate, see elsymrelsrel 38558. Alternate definitions are dfsymrels2 38546, dfsymrels3 38547, dfsymrels4 38548 and dfsymrels5 38549. This definition is similar to the definitions of the classes of reflexive (df-refrels 38512) and transitive (df-trrels 38574) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ SymRels = ( Syms ∩ Rels )
Definition	df-symrel 38545	Define the symmetric relation predicate. (Read: 𝑅 is a symmetric relation.) For sets, being an element of the class of symmetric relations (df-symrels 38544) is equivalent to satisfying the symmetric relation predicate, see elsymrelsrel 38558. Alternate definitions are dfsymrel2 38550 and dfsymrel3 38551. (Contributed by Peter Mazsa, 16-Jul-2021.)
⊢ ( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Theorem	dfsymrels2 38546	Alternate definition of the class of symmetric relations. Cf. the comment of dfrefrels2 38514. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}
Theorem	dfsymrels3 38547*	Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥)}
Theorem	dfsymrels4 38548	Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 = 𝑟}
Theorem	dfsymrels5 38549*	Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}
Theorem	dfsymrel2 38550	Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 19-Apr-2019.) (Revised by Peter Mazsa, 17-Aug-2021.)
⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅))
Theorem	dfsymrel3 38551*	Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 21-Apr-2019.) (Revised by Peter Mazsa, 17-Aug-2021.)
⊢ ( SymRel 𝑅 ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ Rel 𝑅))
Theorem	dfsymrel4 38552	Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ ( SymRel 𝑅 ↔ (◡𝑅 = 𝑅 ∧ Rel 𝑅))
Theorem	dfsymrel5 38553*	Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ ( SymRel 𝑅 ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ Rel 𝑅))
Theorem	elsymrels2 38554	Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))
Theorem	elsymrels3 38555*	Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))
Theorem	elsymrels4 38556	Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 = 𝑅 ∧ 𝑅 ∈ Rels ))
Theorem	elsymrels5 38557*	Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))
Theorem	elsymrelsrel 38558	For sets, being an element of the class of symmetric relations (df-symrels 38544) is equivalent to satisfying the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅))
Theorem	symreleq 38559	Equality theorem for symmetric relation. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.)
⊢ (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆))
Theorem	symrelim 38560	Symmetric relation implies that the domain and the range are equal. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( SymRel 𝑅 → dom 𝑅 = ran 𝑅)
Theorem	symrelcoss 38561	The class of cosets by 𝑅 is symmetric. (Contributed by Peter Mazsa, 20-Dec-2021.)
⊢ SymRel ≀ 𝑅
Theorem	idsymrel 38562	The identity relation is symmetric. (Contributed by AV, 19-Jun-2022.)
⊢ SymRel I
Theorem	epnsymrel 38563	The membership (epsilon) relation is not symmetric. (Contributed by AV, 18-Jun-2022.)
⊢ ¬ SymRel E
21.26.10  Reflexivity and symmetry
Theorem	symrefref2 38564	Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 38565. (Contributed by Peter Mazsa, 19-Jul-2018.)
⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
Theorem	symrefref3 38565*	Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref2 38564. (Contributed by Peter Mazsa, 23-Aug-2021.) (Proof modification is discouraged.)
⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥))
Theorem	refsymrels2 38566	Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels2 38589) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 38514, cf. the comment of dfrefrels2 38514. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
Theorem	refsymrels3 38567*	Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels3 38590) can use the ∀𝑥 ∈ dom 𝑟𝑥𝑟𝑥 version for their reflexive part, not just the ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦) version of dfrefrels3 38515, cf. the comment of dfrefrel3 38517. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.)
⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥))}
Theorem	refsymrel2 38568	A relation which is reflexive and symmetric (like an equivalence relation) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrel2 38516, cf. the comment of dfrefrels2 38514. (Contributed by Peter Mazsa, 23-Aug-2021.)
⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅))
Theorem	refsymrel3 38569*	A relation which is reflexive and symmetric (like an equivalence relation) can use the ∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 version for its reflexive part, not just the ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) version of dfrefrel3 38517, cf. the comment of dfrefrel3 38517. (Contributed by Peter Mazsa, 23-Aug-2021.)
⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ Rel 𝑅))
Theorem	elrefsymrels2 38570	Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels2 38589) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrels2 38514, cf. the comment of dfrefrels2 38514. (Contributed by Peter Mazsa, 22-Jul-2019.)
⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))
Theorem	elrefsymrels3 38571*	Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels3 38590) can use the ∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 version for their reflexive part, not just the ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) version of dfrefrels3 38515, cf. the comment of dfrefrel3 38517. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.)
⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ 𝑅 ∈ Rels ))
Theorem	elrefsymrelsrel 38572	For sets, being an element of the class of reflexive and symmetric relations is equivalent to satisfying the reflexive and symmetric relation predicates. (Contributed by Peter Mazsa, 23-Aug-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ( RefRel 𝑅 ∧ SymRel 𝑅)))
21.26.11  Transitivity
Definition	df-trs 38573	Define the class of all transitive sets (versus the transitive class defined in df-tr 5260). It is used only by df-trrels 38574. Note the similarity of the definitions of df-refs 38511, df-syms 38543 and df-trs 38573. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ Trs = {𝑥 ∣ ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
Definition	df-trrels 38574	Define the class of transitive relations. For sets, being an element of the class of transitive relations is equivalent to satisfying the transitive relation predicate, see eltrrelsrel 38582. Alternate definitions are dftrrels2 38576 and dftrrels3 38577. This definition is similar to the definitions of the classes of reflexive (df-refrels 38512) and symmetric (df-symrels 38544) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ TrRels = ( Trs ∩ Rels )
Definition	df-trrel 38575	Define the transitive relation predicate. (Read: 𝑅 is a transitive relation.) For sets, being an element of the class of transitive relations (df-trrels 38574) is equivalent to satisfying the transitive relation predicate, see eltrrelsrel 38582. Alternate definitions are dftrrel2 38578 and dftrrel3 38579. (Contributed by Peter Mazsa, 17-Jul-2021.)
⊢ ( TrRel 𝑅 ↔ (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Theorem	dftrrels2 38576	Alternate definition of the class of transitive relations. I'd prefer to define the class of transitive relations by using the definition of composition by [Suppes] p. 63. df-coSUP (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝐴𝑢 ∧ 𝑢𝐵𝑦)} as opposed to the present definition of composition df-co 5694 (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝐵𝑢 ∧ 𝑢𝐴𝑦)} because the Suppes definition keeps the order of 𝐴, 𝐵, 𝐶, 𝑅, 𝑆, 𝑇 by default in trsinxpSUP (((𝑅 ∩ (𝐴 × 𝐵)) ∘ (𝑆 ∩ (𝐵 × 𝐶))) ⊆ (𝑇 ∩ (𝐴 × 𝐶)) ↔ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵∀ 𝑧 ∈ 𝐶((𝑥𝑅𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑇𝑧)) while the present definition of composition disarranges them: trsinxp (((𝑆 ∩ (𝐵 × 𝐶)) ∘ (𝑅 ∩ (𝐴 × 𝐵))) ⊆ (𝑇 ∩ (𝐴 × 𝐶 )) ↔ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵∀𝑧 ∈ 𝐶((𝑥𝑅𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑇𝑧) ). This is not mission critical to me, the implication of the Suppes definition is just more aesthetic, at least in the above case. If we swap to the Suppes definition of class composition, I would define the present class of all transitive sets as df-trsSUP and I would consider to switch the definition of the class of cosets by 𝑅 from the present df-coss 38412 to a df-cossSUP. But perhaps there is a mathematical reason to keep the present definition of composition. (Contributed by Peter Mazsa, 21-Jul-2021.)
⊢ TrRels = {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟}
Theorem	dftrrels3 38577*	Alternate definition of the class of transitive relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
⊢ TrRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)}
Theorem	dftrrel2 38578	Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ ( TrRel 𝑅 ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
Theorem	dftrrel3 38579*	Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ ( TrRel 𝑅 ↔ (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ Rel 𝑅))
Theorem	eltrrels2 38580	Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ (𝑅 ∈ TrRels ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))
Theorem	eltrrels3 38581*	Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ (𝑅 ∈ TrRels ↔ (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ 𝑅 ∈ Rels ))
Theorem	eltrrelsrel 38582	For sets, being an element of the class of transitive relations is equivalent to satisfying the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ TrRels ↔ TrRel 𝑅))
Theorem	trreleq 38583	Equality theorem for the transitive relation predicate. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.)
⊢ (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆))
Theorem	trrelressn 38584	Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38443) is transitive. (Contributed by Peter Mazsa, 17-Jun-2024.)
⊢ TrRel (𝑅 ↾ {𝐴})
21.26.12  Equivalence relations
Definition	df-eqvrels 38585	Define the class of equivalence relations. For sets, being an element of the class of equivalence relations is equivalent to satisfying the equivalence relation predicate, see eleqvrelsrel 38595. Alternate definitions are dfeqvrels2 38589 and dfeqvrels3 38590. (Contributed by Peter Mazsa, 7-Nov-2018.)
⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
Definition	df-eqvrel 38586	Define the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.) For sets, being an element of the class of equivalence relations (df-eqvrels 38585) is equivalent to satisfying the equivalence relation predicate, see eleqvrelsrel 38595. Alternate definitions are dfeqvrel2 38591 and dfeqvrel3 38592. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
Definition	df-coeleqvrels 38587	Define the coelement equivalence relations class, the class of sets with coelement equivalence relations. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 38597. Alternate definition is dfcoeleqvrels 38622. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
Definition	df-coeleqvrel 38588	Define the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.) Alternate definition is dfcoeleqvrel 38623. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 38597. (Contributed by Peter Mazsa, 11-Dec-2021.)
⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
Theorem	dfeqvrels2 38589	Alternate definition of the class of equivalence relations. (Contributed by Peter Mazsa, 2-Dec-2019.)
⊢ EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}
Theorem	dfeqvrels3 38590*	Alternate definition of the class of equivalence relations. (Contributed by Peter Mazsa, 2-Dec-2019.)
⊢ EqvRels = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
Theorem	dfeqvrel2 38591	Alternate definition of the equivalence relation predicate. (Contributed by Peter Mazsa, 22-Apr-2019.)
⊢ ( EqvRel 𝑅 ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ Rel 𝑅))
Theorem	dfeqvrel3 38592*	Alternate definition of the equivalence relation predicate. (Contributed by Peter Mazsa, 22-Apr-2019.)
⊢ ( EqvRel 𝑅 ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ Rel 𝑅))
Theorem	eleqvrels2 38593	Element of the class of equivalence relations. (Contributed by Peter Mazsa, 24-Aug-2021.)
⊢ (𝑅 ∈ EqvRels ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))
Theorem	eleqvrels3 38594*	Element of the class of equivalence relations. (Contributed by Peter Mazsa, 24-Aug-2021.)
⊢ (𝑅 ∈ EqvRels ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ 𝑅 ∈ Rels ))
Theorem	eleqvrelsrel 38595	For sets, being an element of the class of equivalence relations is equivalent to satisfying the equivalence relation predicate. (Contributed by Peter Mazsa, 24-Aug-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ EqvRels ↔ EqvRel 𝑅))
Theorem	elcoeleqvrels 38596	Elementhood in the coelement equivalence relations class. (Contributed by Peter Mazsa, 24-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels ))
Theorem	elcoeleqvrelsrel 38597	For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate. (Contributed by Peter Mazsa, 24-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ CoElEqvRel 𝐴))
Theorem	eqvrelrel 38598	An equivalence relation is a relation. (Contributed by Peter Mazsa, 2-Jun-2019.)
⊢ ( EqvRel 𝑅 → Rel 𝑅)
Theorem	eqvrelrefrel 38599	An equivalence relation is reflexive. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → RefRel 𝑅)
Theorem	eqvrelsymrel 38600	An equivalence relation is symmetric. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → SymRel 𝑅)