P. 381 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 38001-38100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cosscnvssid4 38001*	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 38002*	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 38003	Cosets by the empty set are the empty set. (Contributed by Peter Mazsa, 22-Oct-2019.)
⊢ ≀ ∅ = ∅
Theorem	cossid 38004	Cosets by the identity relation are the identity relation. (Contributed by Peter Mazsa, 16-Jan-2019.)
⊢ ≀ I = I
Theorem	cosscnvid 38005	Cosets by the converse identity relation are the identity relation. (Contributed by Peter Mazsa, 27-Sep-2021.)
⊢ ≀ ◡ I = I
Theorem	trcoss 38006*	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 38007	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 38008*	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 38009	Define the relations class. Proper class relations (like I, see reli 5823) are not elements of it. The element of this class and the relation predicate are the same when 𝑅 is a set (see elrelsrel 38011). 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 38011. 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 38034), 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 38009 (see df-refrels 38035 and the resulting dfrefrels2 38037 and dfrefrels3 38038). 3. Finally, in order to be able to work with proper classes (like iprc 7913) as well, we define the predicate of the relation (see df-refrel 38036) so that it is true for the relevant proper classes (see refrelid 38046), and that the element of the class of the required relations (e.g. elrefrels3 38043) and this predicate are the same in case of sets (see elrefrelsrel 38044). (Contributed by Peter Mazsa, 13-Jun-2018.)
⊢ Rels = 𝒫 (V × V)
Theorem	elrels2 38010	The element of the relations class (df-rels 38009) and the relation predicate (df-rel 5680) are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 14-Jun-2018.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
Theorem	elrelsrel 38011	The element of the relations class (df-rels 38009) 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 38012	The element of the relations class is a relation. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ (𝑅 ∈ Rels → Rel 𝑅)
Theorem	elrels5 38013	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 38014	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 38015*	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 38016	Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅))
Theorem	elrelscnveq2 38017*	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 38018*	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 38019	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 38020	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 38021	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 38022*	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 35505 are the same: S = SSet (compare dfssr2 38023 with df-sset 35505), the only reason we do not use dfssr2 38023 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 3958) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, see brssr 38025. Yet in general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for sets, see the comment of df-ss 3958. The only exception (aside from directly investigating the class S e.g. in relssr 38024 or in extssr 38033) is when we have a specific purpose with its usage, like in case of df-refs 38034 versus df-cnvrefs 38049, 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 37834, extep 37807 and extssr 38033, 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 38023	Alternate definition of the subset relation. (Contributed by Peter Mazsa, 9-Aug-2021.)
⊢ S = ((V × V) ∖ ran ( E ⋉ (V ∖ E )))
Theorem	relssr 38024	The subset relation is a relation. (Contributed by Peter Mazsa, 1-Aug-2019.)
⊢ Rel S
Theorem	brssr 38025	The subset relation and subclass relationship (df-ss 3958) 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 38026	Any set is a subset of itself. (Contributed by Peter Mazsa, 1-Aug-2019.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 S 𝐴)
Theorem	issetssr 38027	Two ways of expressing set existence. (Contributed by Peter Mazsa, 1-Aug-2019.)
⊢ (𝐴 ∈ V ↔ 𝐴 S 𝐴)
Theorem	brssrres 38028	Restricted subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)
⊢ (𝐶 ∈ 𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶)))
Theorem	br1cnvssrres 38029	Restricted converse subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)
⊢ (𝐵 ∈ 𝑉 → (𝐵◡( S ↾ 𝐴)𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐵)))
Theorem	brcnvssr 38030	The converse of a subset relation swaps arguments. (Contributed by Peter Mazsa, 1-Aug-2019.)
⊢ (𝐴 ∈ 𝑉 → (𝐴◡ S 𝐵 ↔ 𝐵 ⊆ 𝐴))
Theorem	brcnvssrid 38031	Any set is a converse subset of itself. (Contributed by Peter Mazsa, 9-Jun-2021.)
⊢ (𝐴 ∈ 𝑉 → 𝐴◡ S 𝐴)
Theorem	br1cossxrncnvssrres 38032*	⟨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 38033	Property of subset relation, see also extid 37834, extep 37807 and the comment of df-ssr 38022. (Contributed by Peter Mazsa, 10-Jul-2019.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ S = [𝐵]◡ S ↔ 𝐴 = 𝐵))
21.24.7  Reflexivity
Definition	df-refs 38034	Define the class of all reflexive sets. It is used only by df-refrels 38035. We use subset relation S (df-ssr 38022) here to be able to define converse reflexivity (df-cnvrefs 38049), see also the comment of df-ssr 38022. The elements of this class are not necessarily relations (versus df-refrels 38035). Note the similarity of Definitions df-refs 38034, df-syms 38066 and df-trs 38096, cf. comments of dfrefrels2 38037. (Contributed by Peter Mazsa, 19-Jul-2019.)
⊢ Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
Definition	df-refrels 38035	Define the class of reflexive relations. This is practically dfrefrels2 38037 (which reveals that RefRels can not include proper classes like I as is elements, see comments of dfrefrels2 38037). Another alternative definition is dfrefrels3 38038. The element of this class and the reflexive relation predicate (df-refrel 38036) are the same, that is, (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝐴 is a set, see elrefrelsrel 38044. This definition is similar to the definitions of the classes of symmetric (df-symrels 38067) and transitive (df-trrels 38097) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ RefRels = ( Refs ∩ Rels )
Definition	df-refrel 38036	Define the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.) This is a surprising definition, see the comment of dfrefrel3 38040. Alternate definitions are dfrefrel2 38039 and dfrefrel3 38040. For sets, being an element of the class of reflexive relations (df-refrels 38035) is equivalent to satisfying the reflexive relation predicate, that is (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝑅 is a set, see elrefrelsrel 38044. (Contributed by Peter Mazsa, 16-Jul-2021.)
⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Theorem	dfrefrels2 38037	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 7913) 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 3482. 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 38036. See also the comment of df-rels 38009. Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 38037, it keeps restriction of I: this is why the very similar definitions df-refs 38034, df-syms 38066 and df-trs 38096 diverge when we switch from (general) sets to relations in dfrefrels2 38037, dfsymrels2 38069 and dftrrels2 38099. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
Theorem	dfrefrels3 38038*	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 38039	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 38040*	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 7149 / idrefALT 6113 or df-reflexive 48307 ⊢ (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 38090. Only when this symmetry condition holds, like in case of equivalence relations, see dfeqvrels3 38113, 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 37844 where ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 = 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴𝑥𝑅𝑥). See also similar definition of the converse reflexive relations class dfcnvrefrel3 38055. (Contributed by Peter Mazsa, 8-Jul-2019.)
⊢ ( RefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) ∧ Rel 𝑅))
Theorem	dfrefrel5 38041*	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 38042	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 38043*	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 38044	For sets, being an element of the class of reflexive relations (df-refrels 38035) is equivalent to satisfying the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅))
Theorem	refreleq 38045	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 38046	Identity relation is reflexive. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ RefRel I
Theorem	refrelcoss 38047	The class of cosets by 𝑅 is reflexive. (Contributed by Peter Mazsa, 4-Jul-2020.)
⊢ RefRel ≀ 𝑅
Theorem	refrelressn 38048	Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 37966) is reflexive. (Contributed by Peter Mazsa, 12-Jun-2024.)
⊢ (𝐴 ∈ 𝑉 → RefRel (𝑅 ↾ {𝐴}))
21.24.8  Converse reflexivity
Definition	df-cnvrefs 38049	Define the class of all converse reflexive sets, see the comment of df-ssr 38022. It is used only by df-cnvrefrels 38050. (Contributed by Peter Mazsa, 22-Jul-2019.)
⊢ CnvRefs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥))◡ S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
Definition	df-cnvrefrels 38050	Define the class of converse reflexive relations. This is practically dfcnvrefrels2 38052 (which uses the traditional subclass relation ⊆) : we use converse subset relation (brcnvssr 38030) here to ensure the comparability to the definitions of the classes of all reflexive (df-ref 23422), symmetric (df-syms 38066) and transitive (df-trs 38096) sets. We use this concept to define functions (df-funsALTV 38205, df-funALTV 38206) and disjoints (df-disjs 38228, df-disjALTV 38229). For sets, being an element of the class of converse reflexive relations is equivalent to satisfying the converse reflexive relation predicate, see elcnvrefrelsrel 38060. Alternate definitions are dfcnvrefrels2 38052 and dfcnvrefrels3 38053. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ CnvRefRels = ( CnvRefs ∩ Rels )
Definition	df-cnvrefrel 38051	Define the converse reflexive relation predicate (read: 𝑅 is a converse reflexive relation), see also the comment of dfcnvrefrel3 38055. Alternate definitions are dfcnvrefrel2 38054 and dfcnvrefrel3 38055. (Contributed by Peter Mazsa, 16-Jul-2021.)
⊢ ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Theorem	dfcnvrefrels2 38052	Alternate definition of the class of converse reflexive relations. See the comment of dfrefrels2 38037. (Contributed by Peter Mazsa, 21-Jul-2021.)
⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}
Theorem	dfcnvrefrels3 38053*	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 38054	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 38055*	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 38040. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ ( CnvRefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ Rel 𝑅))
Theorem	dfcnvrefrel4 38056	Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-May-2024.)
⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ I ∧ Rel 𝑅))
Theorem	dfcnvrefrel5 38057*	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 38058	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 38059*	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 38060	For sets, being an element of the class of converse reflexive relations (df-cnvrefrels 38050) is equivalent to satisfying the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅))
Theorem	cnvrefrelcoss2 38061	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 38062	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 38063*	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 38064*	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 38065*	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 ))
21.24.9  Symmetry
Definition	df-syms 38066	Define the class of all symmetric sets. It is used only by df-symrels 38067. Note the similarity of Definitions df-refs 38034, df-syms 38066 and df-trs 38096, cf. the comment of dfrefrels2 38037. (Contributed by Peter Mazsa, 19-Jul-2019.)
⊢ Syms = {𝑥 ∣ ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
Definition	df-symrels 38067	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 38081. Alternate definitions are dfsymrels2 38069, dfsymrels3 38070, dfsymrels4 38071 and dfsymrels5 38072. This definition is similar to the definitions of the classes of reflexive (df-refrels 38035) and transitive (df-trrels 38097) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ SymRels = ( Syms ∩ Rels )
Definition	df-symrel 38068	Define the symmetric relation predicate. (Read: 𝑅 is a symmetric relation.) For sets, being an element of the class of symmetric relations (df-symrels 38067) is equivalent to satisfying the symmetric relation predicate, see elsymrelsrel 38081. Alternate definitions are dfsymrel2 38073 and dfsymrel3 38074. (Contributed by Peter Mazsa, 16-Jul-2021.)
⊢ ( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Theorem	dfsymrels2 38069	Alternate definition of the class of symmetric relations. Cf. the comment of dfrefrels2 38037. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}
Theorem	dfsymrels3 38070*	Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥)}
Theorem	dfsymrels4 38071	Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 = 𝑟}
Theorem	dfsymrels5 38072*	Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}
Theorem	dfsymrel2 38073	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 38074*	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 38075	Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ ( SymRel 𝑅 ↔ (◡𝑅 = 𝑅 ∧ Rel 𝑅))
Theorem	dfsymrel5 38076*	Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ ( SymRel 𝑅 ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ Rel 𝑅))
Theorem	elsymrels2 38077	Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))
Theorem	elsymrels3 38078*	Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))
Theorem	elsymrels4 38079	Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 = 𝑅 ∧ 𝑅 ∈ Rels ))
Theorem	elsymrels5 38080*	Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))
Theorem	elsymrelsrel 38081	For sets, being an element of the class of symmetric relations (df-symrels 38067) is equivalent to satisfying the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅))
Theorem	symreleq 38082	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 38083	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 38084	The class of cosets by 𝑅 is symmetric. (Contributed by Peter Mazsa, 20-Dec-2021.)
⊢ SymRel ≀ 𝑅
Theorem	idsymrel 38085	The identity relation is symmetric. (Contributed by AV, 19-Jun-2022.)
⊢ SymRel I
Theorem	epnsymrel 38086	The membership (epsilon) relation is not symmetric. (Contributed by AV, 18-Jun-2022.)
⊢ ¬ SymRel E
21.24.10  Reflexivity and symmetry
Theorem	symrefref2 38087	Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 38088. (Contributed by Peter Mazsa, 19-Jul-2018.)
⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
Theorem	symrefref3 38088*	Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref2 38087. (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 38089	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 38112) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 38037, cf. the comment of dfrefrels2 38037. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
Theorem	refsymrels3 38090*	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 38113) 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 38038, cf. the comment of dfrefrel3 38040. (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 38091	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 38039, cf. the comment of dfrefrels2 38037. (Contributed by Peter Mazsa, 23-Aug-2021.)
⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅))
Theorem	refsymrel3 38092*	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 38040, cf. the comment of dfrefrel3 38040. (Contributed by Peter Mazsa, 23-Aug-2021.)
⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ Rel 𝑅))
Theorem	elrefsymrels2 38093	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 38112) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrels2 38037, cf. the comment of dfrefrels2 38037. (Contributed by Peter Mazsa, 22-Jul-2019.)
⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))
Theorem	elrefsymrels3 38094*	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 38113) 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 38038, cf. the comment of dfrefrel3 38040. (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 38095	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)))
21.24.11  Transitivity
Definition	df-trs 38096	Define the class of all transitive sets (versus the transitive class defined in df-tr 5262). It is used only by df-trrels 38097. Note the similarity of the definitions of df-refs 38034, df-syms 38066 and df-trs 38096. (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 38097	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 38105. Alternate definitions are dftrrels2 38099 and dftrrels3 38100. This definition is similar to the definitions of the classes of reflexive (df-refrels 38035) and symmetric (df-symrels 38067) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ TrRels = ( Trs ∩ Rels )
Definition	df-trrel 38098	Define the transitive relation predicate. (Read: 𝑅 is a transitive relation.) For sets, being an element of the class of transitive relations (df-trrels 38097) is equivalent to satisfying the transitive relation predicate, see eltrrelsrel 38105. Alternate definitions are dftrrel2 38101 and dftrrel3 38102. (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 38099	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 5682 (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 37935 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 38100*	Alternate definition of the class of transitive relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
⊢ TrRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)}