P. 387 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38601-38700   *Has distinct variable group(s)

Type	Label	Description
Statement
21.26.8  Reflexivity
Definition	df-refs 38601	Define the class of all reflexive sets. It is used only by df-refrels 38602. We use subset relation S (df-ssr 38589) here to be able to define converse reflexivity (df-cnvrefs 38616), see also the comment of df-ssr 38589. The elements of this class are not necessarily relations (versus df-refrels 38602). Note the similarity of Definitions df-refs 38601, df-syms 38633 and df-trs 38667, cf. comments of dfrefrels2 38604. (Contributed by Peter Mazsa, 19-Jul-2019.)
⊢ Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
Definition	df-refrels 38602	Define the class of reflexive relations. This is practically dfrefrels2 38604 (which reveals that RefRels can not include proper classes like I as is elements, see comments of dfrefrels2 38604). Another alternative definition is dfrefrels3 38605. The element of this class and the reflexive relation predicate (df-refrel 38603) are the same, that is, (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝐴 is a set, see elrefrelsrel 38611. This definition is similar to the definitions of the classes of symmetric (df-symrels 38634) and transitive (df-trrels 38668) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ RefRels = ( Refs ∩ Rels )
Definition	df-refrel 38603	Define the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.) This is a surprising definition, see the comment of dfrefrel3 38607. Alternate definitions are dfrefrel2 38606 and dfrefrel3 38607. For sets, being an element of the class of reflexive relations (df-refrels 38602) is equivalent to satisfying the reflexive relation predicate, that is (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝑅 is a set, see elrefrelsrel 38611. (Contributed by Peter Mazsa, 16-Jul-2021.)
⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Theorem	dfrefrels2 38604	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 7841) 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 3457. 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 38603. See also the comment of df-rels 38463. Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 38604, it keeps restriction of I: this is why the very similar definitions df-refs 38601, df-syms 38633 and df-trs 38667 diverge when we switch from (general) sets to relations in dfrefrels2 38604, dfsymrels2 38636 and dftrrels2 38670. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
Theorem	dfrefrels3 38605*	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 38606	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 38607*	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 7079 / idrefALT 6059 or df-reflexive 49868 ⊢ (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 38661. Only when this symmetry condition holds, like in case of equivalence relations, see dfeqvrels3 38684, 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 38357 where ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 = 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴𝑥𝑅𝑥). See also similar definition of the converse reflexive relations class dfcnvrefrel3 38622. (Contributed by Peter Mazsa, 8-Jul-2019.)
⊢ ( RefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) ∧ Rel 𝑅))
Theorem	dfrefrel5 38608*	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 38609	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 38610*	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 38611	For sets, being an element of the class of reflexive relations (df-refrels 38602) is equivalent to satisfying the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))
Theorem	refreleq 38612	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 38613	Identity relation is reflexive. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ RefRel I
Theorem	refrelcoss 38614	The class of cosets by 𝑅 is reflexive. (Contributed by Peter Mazsa, 4-Jul-2020.)
⊢ RefRel ≀ 𝑅
Theorem	refrelressn 38615	Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 38543) is reflexive. (Contributed by Peter Mazsa, 12-Jun-2024.)
⊢ (𝐴 ∈ 𝑉 → RefRel (𝑅 ↾ {𝐴}))
21.26.9  Converse reflexivity
Definition	df-cnvrefs 38616	Define the class of all converse reflexive sets, see the comment of df-ssr 38589. It is used only by df-cnvrefrels 38617. (Contributed by Peter Mazsa, 22-Jul-2019.)
⊢ CnvRefs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥))◡ S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
Definition	df-cnvrefrels 38617	Define the class of converse reflexive relations. This is practically dfcnvrefrels2 38619 (which uses the traditional subclass relation ⊆) : we use converse subset relation (brcnvssr 38597) here to ensure the comparability to the definitions of the classes of all reflexive (df-ref 23420), symmetric (df-syms 38633) and transitive (df-trs 38667) sets. We use this concept to define functions (df-funsALTV 38778, df-funALTV 38779) and disjoints (df-disjs 38801, df-disjALTV 38802). For sets, being an element of the class of converse reflexive relations is equivalent to satisfying the converse reflexive relation predicate, see elcnvrefrelsrel 38627. Alternate definitions are dfcnvrefrels2 38619 and dfcnvrefrels3 38620. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ CnvRefRels = ( CnvRefs ∩ Rels )
Definition	df-cnvrefrel 38618	Define the converse reflexive relation predicate (read: 𝑅 is a converse reflexive relation), see also the comment of dfcnvrefrel3 38622. Alternate definitions are dfcnvrefrel2 38621 and dfcnvrefrel3 38622. (Contributed by Peter Mazsa, 16-Jul-2021.)
⊢ ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Theorem	dfcnvrefrels2 38619	Alternate definition of the class of converse reflexive relations. See the comment of dfrefrels2 38604. (Contributed by Peter Mazsa, 21-Jul-2021.)
⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}
Theorem	dfcnvrefrels3 38620*	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 38621	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 38622*	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 38607. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ ( CnvRefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ Rel 𝑅))
Theorem	dfcnvrefrel4 38623	Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-May-2024.)
⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ I ∧ Rel 𝑅))
Theorem	dfcnvrefrel5 38624*	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 38625	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 38626*	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 38627	For sets, being an element of the class of converse reflexive relations (df-cnvrefrels 38617) is equivalent to satisfying the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅))
Theorem	cnvrefrelcoss2 38628	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 38629	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 38630*	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 38631*	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 38632*	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 ))
21.26.10  Symmetry
Definition	df-syms 38633	Define the class of all symmetric sets. It is used only by df-symrels 38634. Note the similarity of Definitions df-refs 38601, df-syms 38633 and df-trs 38667, cf. the comment of dfrefrels2 38604. (Contributed by Peter Mazsa, 19-Jul-2019.)
⊢ Syms = {𝑥 ∣ ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
Definition	df-symrels 38634	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 38652. Alternate definitions are dfsymrels2 38636, dfsymrels3 38637, dfsymrels4 38642 and dfsymrels5 38643. This definition is similar to the definitions of the classes of reflexive (df-refrels 38602) and transitive (df-trrels 38668) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ SymRels = ( Syms ∩ Rels )
Definition	df-symrel 38635	Define the symmetric relation predicate. (Read: 𝑅 is a symmetric relation.) For sets, being an element of the class of symmetric relations (df-symrels 38634) is equivalent to satisfying the symmetric relation predicate, see elsymrelsrel 38652. Alternate definitions are dfsymrel2 38644 and dfsymrel3 38645. (Contributed by Peter Mazsa, 16-Jul-2021.)
⊢ ( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Theorem	dfsymrels2 38636	Alternate definition of the class of symmetric relations. Cf. the comment of dfrefrels2 38604. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}
Theorem	dfsymrels3 38637*	Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥)}
Theorem	elrelscnveq3 38638*	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 38639	Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅))
Theorem	elrelscnveq2 38640*	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 38641*	Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)))
Theorem	dfsymrels4 38642	Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 = 𝑟}
Theorem	dfsymrels5 38643*	Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}
Theorem	dfsymrel2 38644	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 38645*	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 38646	Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ ( SymRel 𝑅 ↔ (◡𝑅 = 𝑅 ∧ Rel 𝑅))
Theorem	dfsymrel5 38647*	Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ ( SymRel 𝑅 ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ Rel 𝑅))
Theorem	elsymrels2 38648	Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))
Theorem	elsymrels3 38649*	Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))
Theorem	elsymrels4 38650	Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 = 𝑅 ∧ 𝑅 ∈ Rels ))
Theorem	elsymrels5 38651*	Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))
Theorem	elsymrelsrel 38652	For sets, being an element of the class of symmetric relations (df-symrels 38634) is equivalent to satisfying the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅))
Theorem	symreleq 38653	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 38654	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 38655	The class of cosets by 𝑅 is symmetric. (Contributed by Peter Mazsa, 20-Dec-2021.)
⊢ SymRel ≀ 𝑅
Theorem	idsymrel 38656	The identity relation is symmetric. (Contributed by AV, 19-Jun-2022.)
⊢ SymRel I
Theorem	epnsymrel 38657	The membership (epsilon) relation is not symmetric. (Contributed by AV, 18-Jun-2022.)
⊢ ¬ SymRel E
21.26.11  Reflexivity and symmetry
Theorem	symrefref2 38658	Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 38659. (Contributed by Peter Mazsa, 19-Jul-2018.)
⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
Theorem	symrefref3 38659*	Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref2 38658. (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 38660	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 38683) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 38604, cf. the comment of dfrefrels2 38604. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
Theorem	refsymrels3 38661*	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 38684) 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 38605, cf. the comment of dfrefrel3 38607. (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 38662	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 38606, cf. the comment of dfrefrels2 38604. (Contributed by Peter Mazsa, 23-Aug-2021.)
⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅))
Theorem	refsymrel3 38663*	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 38607, cf. the comment of dfrefrel3 38607. (Contributed by Peter Mazsa, 23-Aug-2021.)
⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ Rel 𝑅))
Theorem	elrefsymrels2 38664	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 38683) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrels2 38604, cf. the comment of dfrefrels2 38604. (Contributed by Peter Mazsa, 22-Jul-2019.)
⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))
Theorem	elrefsymrels3 38665*	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 38684) 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 38605, cf. the comment of dfrefrel3 38607. (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 38666	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)))
21.26.12  Transitivity
Definition	df-trs 38667	Define the class of all transitive sets (versus the transitive class defined in df-tr 5197). It is used only by df-trrels 38668. Note the similarity of the definitions of df-refs 38601, df-syms 38633 and df-trs 38667. (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 38668	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 38676. Alternate definitions are dftrrels2 38670 and dftrrels3 38671. This definition is similar to the definitions of the classes of reflexive (df-refrels 38602) and symmetric (df-symrels 38634) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ TrRels = ( Trs ∩ Rels )
Definition	df-trrel 38669	Define the transitive relation predicate. (Read: 𝑅 is a transitive relation.) For sets, being an element of the class of transitive relations (df-trrels 38668) is equivalent to satisfying the transitive relation predicate, see eltrrelsrel 38676. Alternate definitions are dftrrel2 38672 and dftrrel3 38673. (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 38670	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 5623 (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 38512 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 38671*	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 38672	Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ ( TrRel 𝑅 ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
Theorem	dftrrel3 38673*	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 38674	Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ (𝑅 ∈ TrRels ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))
Theorem	eltrrels3 38675*	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 38676	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 38677	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 38678	Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38543) is transitive. (Contributed by Peter Mazsa, 17-Jun-2024.)
⊢ TrRel (𝑅 ↾ {𝐴})
21.26.13  Equivalence relations
Definition	df-eqvrels 38679	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 38689. Alternate definitions are dfeqvrels2 38683 and dfeqvrels3 38684. (Contributed by Peter Mazsa, 7-Nov-2018.)
⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
Definition	df-eqvrel 38680	Define the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.) For sets, being an element of the class of equivalence relations (df-eqvrels 38679) is equivalent to satisfying the equivalence relation predicate, see eleqvrelsrel 38689. Alternate definitions are dfeqvrel2 38685 and dfeqvrel3 38686. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
Definition	df-coeleqvrels 38681	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 38691. Alternate definition is dfcoeleqvrels 38716. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
Definition	df-coeleqvrel 38682	Define the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.) Alternate definition is dfcoeleqvrel 38717. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 38691. (Contributed by Peter Mazsa, 11-Dec-2021.)
⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
Theorem	dfeqvrels2 38683	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 38684*	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 38685	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 38686*	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 38687	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 38688*	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 38689	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 38690	Elementhood in the coelement equivalence relations class. (Contributed by Peter Mazsa, 24-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels ))
Theorem	elcoeleqvrelsrel 38691	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 38692	An equivalence relation is a relation. (Contributed by Peter Mazsa, 2-Jun-2019.)
⊢ ( EqvRel 𝑅 → Rel 𝑅)
Theorem	eqvrelrefrel 38693	An equivalence relation is reflexive. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → RefRel 𝑅)
Theorem	eqvrelsymrel 38694	An equivalence relation is symmetric. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → SymRel 𝑅)
Theorem	eqvreltrrel 38695	An equivalence relation is transitive. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → TrRel 𝑅)
Theorem	eqvrelim 38696	Equivalence relation implies that the domain and the range are equal. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → dom 𝑅 = ran 𝑅)
Theorem	eqvreleq 38697	Equality theorem for equivalence relation. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 23-Sep-2021.)
⊢ (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
Theorem	eqvreleqi 38698	Equality theorem for equivalence relation, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ ( EqvRel 𝑅 ↔ EqvRel 𝑆)
Theorem	eqvreleqd 38699	Equality theorem for equivalence relation, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝜑 → 𝑅 = 𝑆)    ⇒   ⊢ (𝜑 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
Theorem	eqvrelsym 38700	An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.)
⊢ (𝜑 → EqvRel 𝑅)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐵𝑅𝐴)