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Theorem List for Metamath Proof Explorer - 36201-36300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrefrelid 36201 Identity relation is reflexive. (Contributed by Peter Mazsa, 25-Jul-2021.)
RefRel I

Theoremrefrelcoss 36202 The class of cosets by 𝑅 is reflexive. (Contributed by Peter Mazsa, 4-Jul-2020.)
RefRel ≀ 𝑅

20.22.8  Converse reflexivity

Definitiondf-cnvrefs 36203 Define the class of all converse reflexive sets, see the comment of df-ssr 36178. It is used only by df-cnvrefrels 36204. (Contributed by Peter Mazsa, 22-Jul-2019.)
CnvRefs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}

Definitiondf-cnvrefrels 36204 Define the class of converse reflexive relations. This is practically dfcnvrefrels2 36206 (which uses the traditional subclass relation ) : we use converse subset relation (brcnvssr 36186) here to ensure the comparability to the definitions of the classes of all reflexive (df-ref 22205), symmetric (df-syms 36218) and transitive (df-trs 36248) sets.

We use this concept to define functions (df-funsALTV 36354, df-funALTV 36355) and disjoints (df-disjs 36377, df-disjALTV 36378).

For sets, being an element of the class of converse reflexive relations is equivalent to satisfying the converse reflexive relation predicate, see elcnvrefrelsrel 36212. Alternate definitions are dfcnvrefrels2 36206 and dfcnvrefrels3 36207. (Contributed by Peter Mazsa, 7-Jul-2019.)

CnvRefRels = ( CnvRefs ∩ Rels )

Definitiondf-cnvrefrel 36205 Define the converse reflexive relation predicate (read: 𝑅 is a converse reflexive relation), see also the comment of dfcnvrefrel3 36209. Alternate definitions are dfcnvrefrel2 36208 and dfcnvrefrel3 36209. (Contributed by Peter Mazsa, 16-Jul-2021.)
( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))

Theoremdfcnvrefrels2 36206 Alternate definition of the class of converse reflexive relations. See the comment of dfrefrels2 36193. (Contributed by Peter Mazsa, 21-Jul-2021.)
CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}

Theoremdfcnvrefrels3 36207* 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 36208 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 36209* 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 36196. (Contributed by Peter Mazsa, 25-Jul-2021.)
( CnvRefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥𝑅𝑦𝑥 = 𝑦) ∧ Rel 𝑅))

Theoremelcnvrefrels2 36210 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 36211* 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 36212 For sets, being an element of the class of converse reflexive relations (df-cnvrefrels 36204) is equivalent to satisfying the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
(𝑅𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅))

Theoremcnvrefrelcoss2 36213 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 36214 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 36215* 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 36216* 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 36217* 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 ))

20.22.9  Symmetry

Definitiondf-syms 36218 Define the class of all symmetric sets. It is used only by df-symrels 36219.

Note the similarity of Definitions df-refs 36190, df-syms 36218 and df-trs 36248, cf. the comment of dfrefrels2 36193. (Contributed by Peter Mazsa, 19-Jul-2019.)

Syms = {𝑥(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}

Definitiondf-symrels 36219 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 36233. Alternate definitions are dfsymrels2 36221, dfsymrels3 36222, dfsymrels4 36223 and dfsymrels5 36224.

This definition is similar to the definitions of the classes of reflexive (df-refrels 36191) and transitive (df-trrels 36249) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)

SymRels = ( Syms ∩ Rels )

Definitiondf-symrel 36220 Define the symmetric relation predicate. (Read: 𝑅 is a symmetric relation.) For sets, being an element of the class of symmetric relations (df-symrels 36219) is equivalent to satisfying the symmetric relation predicate, see elsymrelsrel 36233. Alternate definitions are dfsymrel2 36225 and dfsymrel3 36226. (Contributed by Peter Mazsa, 16-Jul-2021.)
( SymRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))

Theoremdfsymrels2 36221 Alternate definition of the class of symmetric relations. Cf. the comment of dfrefrels2 36193. (Contributed by Peter Mazsa, 20-Jul-2019.)
SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}

Theoremdfsymrels3 36222* Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
SymRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥)}

Theoremdfsymrels4 36223 Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 20-Jul-2019.)
SymRels = {𝑟 ∈ Rels ∣ 𝑟 = 𝑟}

Theoremdfsymrels5 36224* Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
SymRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥)}

Theoremdfsymrel2 36225 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 36226* 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 36227 Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
( SymRel 𝑅 ↔ (𝑅 = 𝑅 ∧ Rel 𝑅))

Theoremdfsymrel5 36228* Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
( SymRel 𝑅 ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ Rel 𝑅))

Theoremelsymrels2 36229 Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
(𝑅 ∈ SymRels ↔ (𝑅𝑅𝑅 ∈ Rels ))

Theoremelsymrels3 36230* Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
(𝑅 ∈ SymRels ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))

Theoremelsymrels4 36231 Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
(𝑅 ∈ SymRels ↔ (𝑅 = 𝑅𝑅 ∈ Rels ))

Theoremelsymrels5 36232* Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
(𝑅 ∈ SymRels ↔ (∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))

Theoremelsymrelsrel 36233 For sets, being an element of the class of symmetric relations (df-symrels 36219) is equivalent to satisfying the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
(𝑅𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅))

Theoremsymreleq 36234 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 36235 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 36236 The class of cosets by 𝑅 is symmetric. (Contributed by Peter Mazsa, 20-Dec-2021.)
SymRel ≀ 𝑅

Theoremidsymrel 36237 The identity relation is symmetric. (Contributed by AV, 19-Jun-2022.)
SymRel I

Theoremepnsymrel 36238 The membership (epsilon) relation is not symmetric. (Contributed by AV, 18-Jun-2022.)
¬ SymRel E

20.22.10  Reflexivity and symmetry

Theoremsymrefref2 36239 Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 36240. (Contributed by Peter Mazsa, 19-Jul-2018.)
(𝑅𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))

Theoremsymrefref3 36240* Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref2 36239. (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 36241 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 36263) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 36193, cf. the comment of dfrefrels2 36193. (Contributed by Peter Mazsa, 20-Jul-2019.)
( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}

Theoremrefsymrels3 36242* 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 36264) can use the 𝑥 ∈ dom 𝑟𝑥𝑟𝑥 version for their reflexive part, not just the 𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦) version of dfrefrels3 36194, cf. the comment of dfrefrel3 36196. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.)
( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥))}

Theoremrefsymrel2 36243 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 36195, cf. the comment of dfrefrels2 36193. (Contributed by Peter Mazsa, 23-Aug-2021.)
(( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))

Theoremrefsymrel3 36244* 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 36196, cf. the comment of dfrefrel3 36196. (Contributed by Peter Mazsa, 23-Aug-2021.)
(( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ Rel 𝑅))

Theoremelrefsymrels2 36245 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 36263) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrels2 36193, cf. the comment of dfrefrels2 36193. (Contributed by Peter Mazsa, 22-Jul-2019.)
(𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ 𝑅 ∈ Rels ))

Theoremelrefsymrels3 36246* 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 36264) can use the 𝑥 ∈ dom 𝑅𝑥𝑅𝑥 version for their reflexive part, not just the 𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) version of dfrefrels3 36194, cf. the comment of dfrefrel3 36196. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.)
(𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑅 ∈ Rels ))

Theoremelrefsymrelsrel 36247 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)))

20.22.11  Transitivity

Definitiondf-trs 36248 Define the class of all transitive sets (versus the transitive class defined in df-tr 5139). It is used only by df-trrels 36249.

Note the similarity of the definitions of df-refs 36190, df-syms 36218 and df-trs 36248. (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 36249 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 36257. Alternate definitions are dftrrels2 36251 and dftrrels3 36252.

This definition is similar to the definitions of the classes of reflexive (df-refrels 36191) and symmetric (df-symrels 36219) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)

TrRels = ( Trs ∩ Rels )

Definitiondf-trrel 36250 Define the transitive relation predicate. (Read: 𝑅 is a transitive relation.) For sets, being an element of the class of transitive relations (df-trrels 36249) is equivalent to satisfying the transitive relation predicate, see eltrrelsrel 36257. Alternate definitions are dftrrel2 36253 and dftrrel3 36254. (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 36251 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 5533 (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 36099 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 36252* Alternate definition of the class of transitive relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
TrRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)}

Theoremdftrrel2 36253 Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
( TrRel 𝑅 ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅))

Theoremdftrrel3 36254* Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
( TrRel 𝑅 ↔ (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ Rel 𝑅))

Theoremeltrrels2 36255 Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ TrRels ↔ ((𝑅𝑅) ⊆ 𝑅𝑅 ∈ Rels ))

Theoremeltrrels3 36256* Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ TrRels ↔ (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ 𝑅 ∈ Rels ))

Theoremeltrrelsrel 36257 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 36258 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))

20.22.12  Equivalence relations

Definitiondf-eqvrels 36259 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 36269. Alternate definitions are dfeqvrels2 36263 and dfeqvrels3 36264. (Contributed by Peter Mazsa, 7-Nov-2018.)
EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )

Definitiondf-eqvrel 36260 Define the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.) For sets, being an element of the class of equivalence relations (df-eqvrels 36259) is equivalent to satisfying the equivalence relation predicate, see eleqvrelsrel 36269. Alternate definitions are dfeqvrel2 36265 and dfeqvrel3 36266. (Contributed by Peter Mazsa, 17-Apr-2019.)
( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))

Definitiondf-coeleqvrels 36261 Define the 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 36271. Alternate definition is dfcoeleqvrels 36296. (Contributed by Peter Mazsa, 28-Nov-2022.)
CoElEqvRels = {𝑎 ∣ ≀ ( E ↾ 𝑎) ∈ EqvRels }

Definitiondf-coeleqvrel 36262 Define the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.) Alternate definition is dfcoeleqvrel 36297. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 36271. (Contributed by Peter Mazsa, 11-Dec-2021.)
( CoElEqvRel 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))

Theoremdfeqvrels2 36263 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 36264* 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 36265 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 36266* 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))

Theoremeleqvrels2 36267 Element of the class of equivalence relations. (Contributed by Peter Mazsa, 24-Aug-2021.)
(𝑅 ∈ EqvRels ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅 ∧ (𝑅𝑅) ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))

Theoremeleqvrels3 36268* Element of the class of equivalence relations. (Contributed by Peter Mazsa, 24-Aug-2021.)
(𝑅 ∈ EqvRels ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ 𝑅 ∈ Rels ))

Theoremeleqvrelsrel 36269 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.)
(𝑅𝑉 → (𝑅 ∈ EqvRels ↔ EqvRel 𝑅))

Theoremelcoeleqvrels 36270 Elementhood in the coelement equivalence relations class. (Contributed by Peter Mazsa, 24-Jul-2023.)
(𝐴𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ ( E ↾ 𝐴) ∈ EqvRels ))

Theoremelcoeleqvrelsrel 36271 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.)
(𝐴𝑉 → (𝐴 ∈ CoElEqvRels ↔ CoElEqvRel 𝐴))

Theoremeqvrelrel 36272 An equivalence relation is a relation. (Contributed by Peter Mazsa, 2-Jun-2019.)
( EqvRel 𝑅 → Rel 𝑅)

Theoremeqvrelrefrel 36273 An equivalence relation is reflexive. (Contributed by Peter Mazsa, 29-Dec-2021.)
( EqvRel 𝑅 → RefRel 𝑅)

Theoremeqvrelsymrel 36274 An equivalence relation is symmetric. (Contributed by Peter Mazsa, 29-Dec-2021.)
( EqvRel 𝑅 → SymRel 𝑅)

Theoremeqvreltrrel 36275 An equivalence relation is transitive. (Contributed by Peter Mazsa, 29-Dec-2021.)
( EqvRel 𝑅 → TrRel 𝑅)

Theoremeqvrelim 36276 Equivalence relation implies that the domain and the range are equal. (Contributed by Peter Mazsa, 29-Dec-2021.)
( EqvRel 𝑅 → dom 𝑅 = ran 𝑅)

Theoremeqvreleq 36277 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))

Theoremeqvreleqi 36278 Equality theorem for equivalence relation, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)
𝑅 = 𝑆       ( EqvRel 𝑅 ↔ EqvRel 𝑆)

Theoremeqvreleqd 36279 Equality theorem for equivalence relation, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.)
(𝜑𝑅 = 𝑆)       (𝜑 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))

Theoremeqvrelsym 36280 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)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐵𝑅𝐴)

Theoremeqvrelsymb 36281 An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised and distinct variable conditions removed by Peter Mazsa, 2-Jun-2019.)
(𝜑 → EqvRel 𝑅)       (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))

Theoremeqvreltr 36282 An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.)
(𝜑 → EqvRel 𝑅)       (𝜑 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))

Theoremeqvreltrd 36283 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.)
(𝜑 → EqvRel 𝑅)    &   (𝜑𝐴𝑅𝐵)    &   (𝜑𝐵𝑅𝐶)       (𝜑𝐴𝑅𝐶)

Theoremeqvreltr4d 36284 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.)
(𝜑 → EqvRel 𝑅)    &   (𝜑𝐴𝑅𝐵)    &   (𝜑𝐶𝑅𝐵)       (𝜑𝐴𝑅𝐶)

Theoremeqvrelref 36285 An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.)
(𝜑 → EqvRel 𝑅)    &   (𝜑𝐴 ∈ dom 𝑅)       (𝜑𝐴𝑅𝐴)

Theoremeqvrelth 36286 Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (Revised by Peter Mazsa, 2-Jun-2019.)
(𝜑 → EqvRel 𝑅)    &   (𝜑𝐴 ∈ dom 𝑅)       (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Theoremeqvrelcl 36287 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.)
(𝜑 → EqvRel 𝑅)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐴 ∈ dom 𝑅)

Theoremeqvrelthi 36288 Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.)
(𝜑 → EqvRel 𝑅)    &   (𝜑𝐴𝑅𝐵)       (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)

Theoremeqvreldisj 36289 Equivalence classes do not overlap. In other words, two equivalence classes are either equal or disjoint. Theorem 74 of [Suppes] p. 83. (Contributed by NM, 15-Jun-2004.) (Revised by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.)
( EqvRel 𝑅 → ([𝐴]𝑅 = [𝐵]𝑅 ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))

TheoremqsdisjALTV 36290 Elements of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.) (Revised by Peter Mazsa, 3-Jun-2019.)
(𝜑 → EqvRel 𝑅)    &   (𝜑𝐵 ∈ (𝐴 / 𝑅))    &   (𝜑𝐶 ∈ (𝐴 / 𝑅))       (𝜑 → (𝐵 = 𝐶 ∨ (𝐵𝐶) = ∅))

Theoremeqvrelqsel 36291 If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 28-Dec-2019.)
(( EqvRel 𝑅𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶𝐵) → 𝐵 = [𝐶]𝑅)

Theoremeqvrelcoss 36292 Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 4-Jul-2020.) (Revised by Peter Mazsa, 20-Dec-2021.)
( EqvRel ≀ 𝑅 ↔ TrRel ≀ 𝑅)

Theoremeqvrelcoss3 36293* Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 28-Apr-2019.)
( EqvRel ≀ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Theoremeqvrelcoss2 36294 Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019.)
( EqvRel ≀ 𝑅 ↔ ≀ ≀ 𝑅 ⊆ ≀ 𝑅)

Theoremeqvrelcoss4 36295* Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019.) (Revised by Peter Mazsa, 30-Sep-2021.)
( EqvRel ≀ 𝑅 ↔ ∀𝑥𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))

Theoremdfcoeleqvrels 36296 Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 36294, eqvrelcoss3 36293 and eqvrelcoss4 36295 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)
CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }

Theoremdfcoeleqvrel 36297 Alternate definition of the coelement equivalence relation predicate: a coelement equivalence relation is an equivalence relation on coelements. Other alternate definitions should be based on eqvrelcoss2 36294, eqvrelcoss3 36293 and eqvrelcoss4 36295 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)
( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)

20.22.13  Redundancy

Definitiondf-redunds 36298* Define the class of all redundant sets 𝑥 with respect to 𝑦 in 𝑧. For sets, binary relation on the class of all redundant sets (brredunds 36301) is equivalent to satisfying the redundancy predicate (df-redund 36299). (Contributed by Peter Mazsa, 23-Oct-2022.)
Redunds = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥𝑦 ∧ (𝑥𝑧) = (𝑦𝑧))}

Definitiondf-redund 36299 Define the redundancy predicate. Read: 𝐴 is redundant with respect to 𝐵 in 𝐶. For sets, binary relation on the class of all redundant sets (brredunds 36301) is equivalent to satisfying the redundancy predicate. (Contributed by Peter Mazsa, 23-Oct-2022.)
(𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ (𝐴𝐵 ∧ (𝐴𝐶) = (𝐵𝐶)))

Definitiondf-redundp 36300 Define the redundancy operator for propositions, cf. df-redund 36299. (Contributed by Peter Mazsa, 23-Oct-2022.)
( redund (𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ ((𝜑𝜒) ↔ (𝜓𝜒))))

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