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Theorem List for Metamath Proof Explorer - 38201-38300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelrelscnveq 38201 Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ Rels → (𝑅𝑅𝑅 = 𝑅))
 
Theoremelrelscnveq2 38202* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ Rels → (𝑅 = 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
 
Theoremelrelscnveq4 38203* Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ Rels → (𝑅𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)))
 
Theoremcnvelrels 38204 The converse of a set is an element of the class of relations. (Contributed by Peter Mazsa, 18-Aug-2019.)
(𝐴𝑉𝐴 ∈ Rels )
 
Theoremcosselrels 38205 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 38206 Cosets of converse sets are elements of the relations class. (Contributed by Peter Mazsa, 31-Aug-2021.)
(𝐴𝑉 → ≀ 𝐴 ∈ Rels )
 
21.25.6  Subset relations
 
Definitiondf-ssr 38207* Define the subsets class or the class of subset relations. Similar to definitions of epsilon relation (df-eprel 5577) and identity relation (df-id 5571) classes. Subset relation class and Scott Fenton's subset class df-sset 35691 are the same: S = SSet (compare dfssr2 38208 with df-sset 35691), the only reason we do not use dfssr2 38208 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 3964) are the same, that is, (𝐴 S 𝐵𝐴𝐵) when 𝐵 is a set, see brssr 38210. Yet in general we use the subclass relation 𝐴𝐵 both for classes and for sets, see the comment of df-ss 3964. The only exception (aside from directly investigating the class S e.g. in relssr 38209 or in extssr 38218) is when we have a specific purpose with its usage, like in case of df-refs 38219 versus df-cnvrefs 38234, 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 38019, extep 37992 and extssr 38218, 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 38208 Alternate definition of the subset relation. (Contributed by Peter Mazsa, 9-Aug-2021.)
S = ((V × V) ∖ ran ( E ⋉ (V ∖ E )))
 
Theoremrelssr 38209 The subset relation is a relation. (Contributed by Peter Mazsa, 1-Aug-2019.)
Rel S
 
Theorembrssr 38210 The subset relation and subclass relationship (df-ss 3964) 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 38211 Any set is a subset of itself. (Contributed by Peter Mazsa, 1-Aug-2019.)
(𝐴𝑉𝐴 S 𝐴)
 
Theoremissetssr 38212 Two ways of expressing set existence. (Contributed by Peter Mazsa, 1-Aug-2019.)
(𝐴 ∈ V ↔ 𝐴 S 𝐴)
 
Theorembrssrres 38213 Restricted subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)
(𝐶𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐵𝐶)))
 
Theorembr1cnvssrres 38214 Restricted converse subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)
(𝐵𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐶𝐴𝐶𝐵)))
 
Theorembrcnvssr 38215 The converse of a subset relation swaps arguments. (Contributed by Peter Mazsa, 1-Aug-2019.)
(𝐴𝑉 → (𝐴 S 𝐵𝐵𝐴))
 
Theorembrcnvssrid 38216 Any set is a converse subset of itself. (Contributed by Peter Mazsa, 9-Jun-2021.)
(𝐴𝑉𝐴 S 𝐴)
 
Theorembr1cossxrncnvssrres 38217* 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 38218 Property of subset relation, see also extid 38019, extep 37992 and the comment of df-ssr 38207. (Contributed by Peter Mazsa, 10-Jul-2019.)
((𝐴𝑉𝐵𝑊) → ([𝐴] S = [𝐵] S ↔ 𝐴 = 𝐵))
 
21.25.7  Reflexivity
 
Definitiondf-refs 38219 Define the class of all reflexive sets. It is used only by df-refrels 38220. We use subset relation S (df-ssr 38207) here to be able to define converse reflexivity (df-cnvrefs 38234), see also the comment of df-ssr 38207. The elements of this class are not necessarily relations (versus df-refrels 38220).

Note the similarity of Definitions df-refs 38219, df-syms 38251 and df-trs 38281, cf. comments of dfrefrels2 38222. (Contributed by Peter Mazsa, 19-Jul-2019.)

Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
 
Definitiondf-refrels 38220 Define the class of reflexive relations. This is practically dfrefrels2 38222 (which reveals that RefRels can not include proper classes like I as is elements, see comments of dfrefrels2 38222).

Another alternative definition is dfrefrels3 38223. The element of this class and the reflexive relation predicate (df-refrel 38221) are the same, that is, (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝐴 is a set, see elrefrelsrel 38229.

This definition is similar to the definitions of the classes of symmetric (df-symrels 38252) and transitive (df-trrels 38282) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)

RefRels = ( Refs ∩ Rels )
 
Definitiondf-refrel 38221 Define the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.) This is a surprising definition, see the comment of dfrefrel3 38225. Alternate definitions are dfrefrel2 38224 and dfrefrel3 38225. For sets, being an element of the class of reflexive relations (df-refrels 38220) is equivalent to satisfying the reflexive relation predicate, that is (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝑅 is a set, see elrefrelsrel 38229. (Contributed by Peter Mazsa, 16-Jul-2021.)
( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
 
Theoremdfrefrels2 38222 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 7914) 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 3483. 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 38221. See also the comment of df-rels 38194.

Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 38222, it keeps restriction of I: this is why the very similar definitions df-refs 38219, df-syms 38251 and df-trs 38281 diverge when we switch from (general) sets to relations in dfrefrels2 38222, dfsymrels2 38254 and dftrrels2 38284. (Contributed by Peter Mazsa, 20-Jul-2019.)

RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
 
Theoremdfrefrels3 38223* 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 38224 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 38225* 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 7150 / idrefALT 6114 or df-reflexive 48548 (𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥𝐴𝑥𝑅𝑥)). It turns out that the not-surprising definition which contains 𝑥 ∈ dom 𝑟𝑥𝑟𝑥 needs symmetry as well, see refsymrels3 38275. Only when this symmetry condition holds, like in case of equivalence relations, see dfeqvrels3 38298, 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 38029 where (∀𝑥𝐴𝑦𝐴(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥𝐴𝑥𝑅𝑥). See also similar definition of the converse reflexive relations class dfcnvrefrel3 38240. (Contributed by Peter Mazsa, 8-Jul-2019.)

( RefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ∧ Rel 𝑅))
 
Theoremdfrefrel5 38226* Alternate definition of the reflexive relation predicate. (Contributed by Peter Mazsa, 12-Dec-2023.)
( RefRel 𝑅 ↔ (∀𝑥 ∈ (dom 𝑅 ∩ ran 𝑅)𝑥𝑅𝑥 ∧ Rel 𝑅))
 
Theoremelrefrels2 38227 Element of the class of reflexive relations. (Contributed by Peter Mazsa, 23-Jul-2019.)
(𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅𝑅 ∈ Rels ))
 
Theoremelrefrels3 38228* 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 38229 For sets, being an element of the class of reflexive relations (df-refrels 38220) is equivalent to satisfying the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
(𝑅𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))
 
Theoremrefreleq 38230 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 38231 Identity relation is reflexive. (Contributed by Peter Mazsa, 25-Jul-2021.)
RefRel I
 
Theoremrefrelcoss 38232 The class of cosets by 𝑅 is reflexive. (Contributed by Peter Mazsa, 4-Jul-2020.)
RefRel ≀ 𝑅
 
Theoremrefrelressn 38233 Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 38151) is reflexive. (Contributed by Peter Mazsa, 12-Jun-2024.)
(𝐴𝑉 → RefRel (𝑅 ↾ {𝐴}))
 
21.25.8  Converse reflexivity
 
Definitiondf-cnvrefs 38234 Define the class of all converse reflexive sets, see the comment of df-ssr 38207. It is used only by df-cnvrefrels 38235. (Contributed by Peter Mazsa, 22-Jul-2019.)
CnvRefs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
 
Definitiondf-cnvrefrels 38235 Define the class of converse reflexive relations. This is practically dfcnvrefrels2 38237 (which uses the traditional subclass relation ) : we use converse subset relation (brcnvssr 38215) here to ensure the comparability to the definitions of the classes of all reflexive (df-ref 23495), symmetric (df-syms 38251) and transitive (df-trs 38281) sets.

We use this concept to define functions (df-funsALTV 38390, df-funALTV 38391) and disjoints (df-disjs 38413, df-disjALTV 38414).

For sets, being an element of the class of converse reflexive relations is equivalent to satisfying the converse reflexive relation predicate, see elcnvrefrelsrel 38245. Alternate definitions are dfcnvrefrels2 38237 and dfcnvrefrels3 38238. (Contributed by Peter Mazsa, 7-Jul-2019.)

CnvRefRels = ( CnvRefs ∩ Rels )
 
Definitiondf-cnvrefrel 38236 Define the converse reflexive relation predicate (read: 𝑅 is a converse reflexive relation), see also the comment of dfcnvrefrel3 38240. Alternate definitions are dfcnvrefrel2 38239 and dfcnvrefrel3 38240. (Contributed by Peter Mazsa, 16-Jul-2021.)
( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
 
Theoremdfcnvrefrels2 38237 Alternate definition of the class of converse reflexive relations. See the comment of dfrefrels2 38222. (Contributed by Peter Mazsa, 21-Jul-2021.)
CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}
 
Theoremdfcnvrefrels3 38238* Alternate definition of the class of converse reflexive relations. (Contributed by Peter Mazsa, 22-Jul-2019.)
CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦)}
 
Theoremdfcnvrefrel2 38239 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))
 
Theoremdfcnvrefrel3 38240* 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 38225. (Contributed by Peter Mazsa, 25-Jul-2021.)
( CnvRefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥𝑅𝑦𝑥 = 𝑦) ∧ Rel 𝑅))
 
Theoremdfcnvrefrel4 38241 Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-May-2024.)
( CnvRefRel 𝑅 ↔ (𝑅 ⊆ I ∧ Rel 𝑅))
 
Theoremdfcnvrefrel5 38242* Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-May-2024.)
( CnvRefRel 𝑅 ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑥 = 𝑦) ∧ Rel 𝑅))
 
Theoremelcnvrefrels2 38243 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 ))
 
Theoremelcnvrefrels3 38244* Element of the class of converse reflexive relations. (Contributed by Peter Mazsa, 30-Aug-2021.)
(𝑅 ∈ CnvRefRels ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥𝑅𝑦𝑥 = 𝑦) ∧ 𝑅 ∈ Rels ))
 
Theoremelcnvrefrelsrel 38245 For sets, being an element of the class of converse reflexive relations (df-cnvrefrels 38235) is equivalent to satisfying the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
(𝑅𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅))
 
Theoremcnvrefrelcoss2 38246 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 )
 
Theoremcosselcnvrefrels2 38247 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 ))
 
Theoremcosselcnvrefrels3 38248* 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 ↔ (∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝑅 ∈ Rels ))
 
Theoremcosselcnvrefrels4 38249* 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 ))
 
Theoremcosselcnvrefrels5 38250* 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 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ∧ ≀ 𝑅 ∈ Rels ))
 
21.25.9  Symmetry
 
Definitiondf-syms 38251 Define the class of all symmetric sets. It is used only by df-symrels 38252.

Note the similarity of Definitions df-refs 38219, df-syms 38251 and df-trs 38281, cf. the comment of dfrefrels2 38222. (Contributed by Peter Mazsa, 19-Jul-2019.)

Syms = {𝑥(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
 
Definitiondf-symrels 38252 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 38266. Alternate definitions are dfsymrels2 38254, dfsymrels3 38255, dfsymrels4 38256 and dfsymrels5 38257.

This definition is similar to the definitions of the classes of reflexive (df-refrels 38220) and transitive (df-trrels 38282) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)

SymRels = ( Syms ∩ Rels )
 
Definitiondf-symrel 38253 Define the symmetric relation predicate. (Read: 𝑅 is a symmetric relation.) For sets, being an element of the class of symmetric relations (df-symrels 38252) is equivalent to satisfying the symmetric relation predicate, see elsymrelsrel 38266. Alternate definitions are dfsymrel2 38258 and dfsymrel3 38259. (Contributed by Peter Mazsa, 16-Jul-2021.)
( SymRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
 
Theoremdfsymrels2 38254 Alternate definition of the class of symmetric relations. Cf. the comment of dfrefrels2 38222. (Contributed by Peter Mazsa, 20-Jul-2019.)
SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
 
Theoremdfsymrels3 38255* Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
SymRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥)}
 
Theoremdfsymrels4 38256 Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 20-Jul-2019.)
SymRels = {𝑟 ∈ Rels ∣ 𝑟 = 𝑟}
 
Theoremdfsymrels5 38257* Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
SymRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥)}
 
Theoremdfsymrel2 38258 Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 19-Apr-2019.) (Revised by Peter Mazsa, 17-Aug-2021.)
( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
 
Theoremdfsymrel3 38259* Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 21-Apr-2019.) (Revised by Peter Mazsa, 17-Aug-2021.)
( SymRel 𝑅 ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel 𝑅))
 
Theoremdfsymrel4 38260 Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
( SymRel 𝑅 ↔ (𝑅 = 𝑅 ∧ Rel 𝑅))
 
Theoremdfsymrel5 38261* Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
( SymRel 𝑅 ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel 𝑅))
 
Theoremelsymrels2 38262 Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
(𝑅 ∈ SymRels ↔ (𝑅𝑅𝑅 ∈ Rels ))
 
Theoremelsymrels3 38263* Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
(𝑅 ∈ SymRels ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))
 
Theoremelsymrels4 38264 Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
(𝑅 ∈ SymRels ↔ (𝑅 = 𝑅𝑅 ∈ Rels ))
 
Theoremelsymrels5 38265* Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
(𝑅 ∈ SymRels ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))
 
Theoremelsymrelsrel 38266 For sets, being an element of the class of symmetric relations (df-symrels 38252) is equivalent to satisfying the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
(𝑅𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅))
 
Theoremsymreleq 38267 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))
 
Theoremsymrelim 38268 Symmetric relation implies that the domain and the range are equal. (Contributed by Peter Mazsa, 29-Dec-2021.)
( SymRel 𝑅 → dom 𝑅 = ran 𝑅)
 
Theoremsymrelcoss 38269 The class of cosets by 𝑅 is symmetric. (Contributed by Peter Mazsa, 20-Dec-2021.)
SymRel ≀ 𝑅
 
Theoremidsymrel 38270 The identity relation is symmetric. (Contributed by AV, 19-Jun-2022.)
SymRel I
 
Theoremepnsymrel 38271 The membership (epsilon) relation is not symmetric. (Contributed by AV, 18-Jun-2022.)
¬ SymRel E
 
21.25.10  Reflexivity and symmetry
 
Theoremsymrefref2 38272 Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 38273. (Contributed by Peter Mazsa, 19-Jul-2018.)
(𝑅𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
 
Theoremsymrefref3 38273* Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref2 38272. (Contributed by Peter Mazsa, 23-Aug-2021.) (Proof modification is discouraged.)
(∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) → (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥))
 
Theoremrefsymrels2 38274 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 38297) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 38222, cf. the comment of dfrefrels2 38222. (Contributed by Peter Mazsa, 20-Jul-2019.)
( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
 
Theoremrefsymrels3 38275* 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 38298) can use the 𝑥 ∈ dom 𝑟𝑥𝑟𝑥 version for their reflexive part, not just the 𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦) version of dfrefrels3 38223, cf. the comment of dfrefrel3 38225. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.)
( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥))}
 
Theoremrefsymrel2 38276 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 38224, cf. the comment of dfrefrels2 38222. (Contributed by Peter Mazsa, 23-Aug-2021.)
(( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
 
Theoremrefsymrel3 38277* 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 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) version of dfrefrel3 38225, cf. the comment of dfrefrel3 38225. (Contributed by Peter Mazsa, 23-Aug-2021.)
(( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ Rel 𝑅))
 
Theoremelrefsymrels2 38278 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 38297) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrels2 38222, cf. the comment of dfrefrels2 38222. (Contributed by Peter Mazsa, 22-Jul-2019.)
(𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ 𝑅 ∈ Rels ))
 
Theoremelrefsymrels3 38279* 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 38298) can use the 𝑥 ∈ dom 𝑅𝑥𝑅𝑥 version for their reflexive part, not just the 𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) version of dfrefrels3 38223, cf. the comment of dfrefrel3 38225. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.)
(𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑅 ∈ Rels ))
 
Theoremelrefsymrelsrel 38280 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.)
(𝑅𝑉 → (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ( RefRel 𝑅 ∧ SymRel 𝑅)))
 
21.25.11  Transitivity
 
Definitiondf-trs 38281 Define the class of all transitive sets (versus the transitive class defined in df-tr 5262). It is used only by df-trrels 38282.

Note the similarity of the definitions of df-refs 38219, df-syms 38251 and df-trs 38281. (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))}
 
Definitiondf-trrels 38282 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 38290. Alternate definitions are dftrrels2 38284 and dftrrels3 38285.

This definition is similar to the definitions of the classes of reflexive (df-refrels 38220) and symmetric (df-symrels 38252) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)

TrRels = ( Trs ∩ Rels )
 
Definitiondf-trrel 38283 Define the transitive relation predicate. (Read: 𝑅 is a transitive relation.) For sets, being an element of the class of transitive relations (df-trrels 38282) is equivalent to satisfying the transitive relation predicate, see eltrrelsrel 38290. Alternate definitions are dftrrel2 38286 and dftrrel3 38287. (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))
 
Theoremdftrrels2 38284 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 (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝐴𝑢𝑢𝐵𝑦)} as opposed to the present definition of composition df-co 5682 (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝐵𝑢𝑢𝐴𝑦)} 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)) ↔ ∀𝑥𝐴𝑦𝐵 𝑧𝐶((𝑥𝑅𝑦𝑦𝑆𝑧) → 𝑥𝑇𝑧)) while the present definition of composition disarranges them: trsinxp (((𝑆 ∩ (𝐵 × 𝐶)) ∘ (𝑅 ∩ (𝐴 × 𝐵))) ⊆ (𝑇 ∩ (𝐴 × 𝐶 )) ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶((𝑥𝑅𝑦𝑦𝑆𝑧) → 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 38120 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 ∣ (𝑟𝑟) ⊆ 𝑟}
 
Theoremdftrrels3 38285* Alternate definition of the class of transitive relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
TrRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)}
 
Theoremdftrrel2 38286 Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
( TrRel 𝑅 ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
 
Theoremdftrrel3 38287* Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
( TrRel 𝑅 ↔ (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ Rel 𝑅))
 
Theoremeltrrels2 38288 Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ TrRels ↔ ((𝑅𝑅) ⊆ 𝑅𝑅 ∈ Rels ))
 
Theoremeltrrels3 38289* Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ TrRels ↔ (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ 𝑅 ∈ Rels ))
 
Theoremeltrrelsrel 38290 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.)
(𝑅𝑉 → (𝑅 ∈ TrRels ↔ TrRel 𝑅))
 
Theoremtrreleq 38291 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))
 
Theoremtrrelressn 38292 Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 38151) is transitive. (Contributed by Peter Mazsa, 17-Jun-2024.)
TrRel (𝑅 ↾ {𝐴})
 
21.25.12  Equivalence relations
 
Definitiondf-eqvrels 38293 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 38303. Alternate definitions are dfeqvrels2 38297 and dfeqvrels3 38298. (Contributed by Peter Mazsa, 7-Nov-2018.)
EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
 
Definitiondf-eqvrel 38294 Define the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.) For sets, being an element of the class of equivalence relations (df-eqvrels 38293) is equivalent to satisfying the equivalence relation predicate, see eleqvrelsrel 38303. Alternate definitions are dfeqvrel2 38299 and dfeqvrel3 38300. (Contributed by Peter Mazsa, 17-Apr-2019.)
( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
 
Definitiondf-coeleqvrels 38295 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 38305. Alternate definition is dfcoeleqvrels 38330. (Contributed by Peter Mazsa, 28-Nov-2022.)
CoElEqvRels = {𝑎 ∣ ≀ ( E ↾ 𝑎) ∈ EqvRels }
 
Definitiondf-coeleqvrel 38296 Define the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.) Alternate definition is dfcoeleqvrel 38331. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 38305. (Contributed by Peter Mazsa, 11-Dec-2021.)
( CoElEqvRel 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
 
Theoremdfeqvrels2 38297 Alternate definition of the class of equivalence relations. (Contributed by Peter Mazsa, 2-Dec-2019.)
EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}
 
Theoremdfeqvrels3 38298* Alternate definition of the class of equivalence relations. (Contributed by Peter Mazsa, 2-Dec-2019.)
EqvRels = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))}
 
Theoremdfeqvrel2 38299 Alternate definition of the equivalence relation predicate. (Contributed by Peter Mazsa, 22-Apr-2019.)
( EqvRel 𝑅 ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅) ∧ Rel 𝑅))
 
Theoremdfeqvrel3 38300* Alternate definition of the equivalence relation predicate. (Contributed by Peter Mazsa, 22-Apr-2019.)
( EqvRel 𝑅 ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ Rel 𝑅))
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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48500 486 48501-48587
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