P. 394 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39301-39400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dfcnvrefrel3 39301*	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 39286. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ ( CnvRefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ Rel 𝑅))
Theorem	dfcnvrefrel4 39302	Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-May-2024.)
⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ I ∧ Rel 𝑅))
Theorem	dfcnvrefrel5 39303*	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 39304	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 39305*	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 39306	For sets, being an element of the class of converse reflexive relations (df-cnvrefrels 39296) is equivalent to satisfying the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅))
Theorem	cnvrefrelcoss2 39307	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 39308	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 39309*	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 39310*	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 39311*	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 ))
21.26.11  Symmetry
Definition	df-syms 39312	Define the class of all symmetric sets. It is used only by df-symrels 39313. Note the similarity of Definitions df-refs 39280, df-syms 39312 and df-trs 39346, cf. the comment of dfrefrels2 39283. (Contributed by Peter Mazsa, 19-Jul-2019.)
⊢ Syms = {𝑥 ∣ ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))}
Definition	df-symrels 39313	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 39331. Alternate definitions are dfsymrels2 39315, dfsymrels3 39316, dfsymrels4 39321 and dfsymrels5 39322. This definition is similar to the definitions of the classes of reflexive (df-refrels 39281) and transitive (df-trrels 39347) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ SymRels = ( Syms ∩ Rels )
Definition	df-symrel 39314	Define the symmetric relation predicate. (Read: 𝑅 is a symmetric relation.) For sets, being an element of the class of symmetric relations (df-symrels 39313) is equivalent to satisfying the symmetric relation predicate, see elsymrelsrel 39331. Alternate definitions are dfsymrel2 39323 and dfsymrel3 39324. (Contributed by Peter Mazsa, 16-Jul-2021.)
⊢ ( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Theorem	dfsymrels2 39315	Alternate definition of the class of symmetric relations. Cf. the comment of dfrefrels2 39283. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}
Theorem	dfsymrels3 39316*	Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥)}
Theorem	elrelscnveq3 39317*	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 39318	Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅))
Theorem	elrelscnveq2 39319*	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 39320*	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 39321	Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 = 𝑟}
Theorem	dfsymrels5 39322*	Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}
Theorem	dfsymrel2 39323	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 39324*	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 39325	Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ ( SymRel 𝑅 ↔ (◡𝑅 = 𝑅 ∧ Rel 𝑅))
Theorem	dfsymrel5 39326*	Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ ( SymRel 𝑅 ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ Rel 𝑅))
Theorem	elsymrels2 39327	Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))
Theorem	elsymrels3 39328*	Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))
Theorem	elsymrels4 39329	Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 = 𝑅 ∧ 𝑅 ∈ Rels ))
Theorem	elsymrels5 39330*	Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels ))
Theorem	elsymrelsrel 39331	For sets, being an element of the class of symmetric relations (df-symrels 39313) is equivalent to satisfying the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅))
Theorem	symreleq 39332	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 39333	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 39334	The class of cosets by 𝑅 is symmetric. (Contributed by Peter Mazsa, 20-Dec-2021.)
⊢ SymRel ≀ 𝑅
Theorem	idsymrel 39335	The identity relation is symmetric. (Contributed by AV, 19-Jun-2022.)
⊢ SymRel I
Theorem	epnsymrel 39336	The membership (epsilon) relation is not symmetric. (Contributed by AV, 18-Jun-2022.)
⊢ ¬ SymRel E
21.26.12  Reflexivity and symmetry
Theorem	symrefref2 39337	Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 39338. (Contributed by Peter Mazsa, 19-Jul-2018.)
⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
Theorem	symrefref3 39338*	Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref2 39337. (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 39339	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 39362) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 39283, cf. the comment of dfrefrels2 39283. (Contributed by Peter Mazsa, 20-Jul-2019.)
⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)}
Theorem	refsymrels3 39340*	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 39363) 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 39284, cf. the comment of dfrefrel3 39286. (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 39341	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 39285, cf. the comment of dfrefrels2 39283. (Contributed by Peter Mazsa, 23-Aug-2021.)
⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅))
Theorem	refsymrel3 39342*	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 39286, cf. the comment of dfrefrel3 39286. (Contributed by Peter Mazsa, 23-Aug-2021.)
⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ Rel 𝑅))
Theorem	elrefsymrels2 39343	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 39362) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrels2 39283, cf. the comment of dfrefrels2 39283. (Contributed by Peter Mazsa, 22-Jul-2019.)
⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ 𝑅 ∈ Rels ))
Theorem	elrefsymrels3 39344*	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 39363) 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 39284, cf. the comment of dfrefrel3 39286. (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 39345	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)))
21.26.13  Transitivity
Definition	df-trs 39346	Define the class of all transitive sets (versus the transitive class defined in df-tr 5224). It is used only by df-trrels 39347. Note the similarity of the definitions of df-refs 39280, df-syms 39312 and df-trs 39346. (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 39347	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 39355. Alternate definitions are dftrrels2 39349 and dftrrels3 39350. This definition is similar to the definitions of the classes of reflexive (df-refrels 39281) and symmetric (df-symrels 39313) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)
⊢ TrRels = ( Trs ∩ Rels )
Definition	df-trrel 39348	Define the transitive relation predicate. (Read: 𝑅 is a transitive relation.) For sets, being an element of the class of transitive relations (df-trrels 39347) is equivalent to satisfying the transitive relation predicate, see eltrrelsrel 39355. Alternate definitions are dftrrel2 39351 and dftrrel3 39352. (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 39349	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 5675 (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 39191 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 39350*	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 39351	Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ ( TrRel 𝑅 ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
Theorem	dftrrel3 39352*	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 39353	Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)
⊢ (𝑅 ∈ TrRels ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ 𝑅 ∈ Rels ))
Theorem	eltrrels3 39354*	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 39355	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 39356	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 39357	Any class ' R ' restricted to the singleton of the class ' A ' (see ressn2 39222) is transitive. (Contributed by Peter Mazsa, 17-Jun-2024.)
⊢ TrRel (𝑅 ↾ {𝐴})
21.26.14  Equivalence relations
Definition	df-eqvrels 39358	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 39368. Alternate definitions are dfeqvrels2 39362 and dfeqvrels3 39363. (Contributed by Peter Mazsa, 7-Nov-2018.)
⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
Definition	df-eqvrel 39359	Define the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.) For sets, being an element of the class of equivalence relations (df-eqvrels 39358) is equivalent to satisfying the equivalence relation predicate, see eleqvrelsrel 39368. Alternate definitions are dfeqvrel2 39364 and dfeqvrel3 39365. (Contributed by Peter Mazsa, 17-Apr-2019.)
⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
Definition	df-coeleqvrels 39360	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 39370. Alternate definition is dfcoeleqvrels 39395. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
Definition	df-coeleqvrel 39361	Define the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.) Alternate definition is dfcoeleqvrel 39396. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 39370. (Contributed by Peter Mazsa, 11-Dec-2021.)
⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
Theorem	dfeqvrels2 39362	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 39363*	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 39364	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 39365*	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 39366	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 39367*	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 39368	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 39369	Elementhood in the coelement equivalence relations class. (Contributed by Peter Mazsa, 24-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels ))
Theorem	elcoeleqvrelsrel 39370	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 39371	An equivalence relation is a relation. (Contributed by Peter Mazsa, 2-Jun-2019.)
⊢ ( EqvRel 𝑅 → Rel 𝑅)
Theorem	eqvrelrefrel 39372	An equivalence relation is reflexive. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → RefRel 𝑅)
Theorem	eqvrelsymrel 39373	An equivalence relation is symmetric. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → SymRel 𝑅)
Theorem	eqvreltrrel 39374	An equivalence relation is transitive. (Contributed by Peter Mazsa, 29-Dec-2021.)
⊢ ( EqvRel 𝑅 → TrRel 𝑅)
Theorem	eqvrelim 39375	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 39376	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 39377	Equality theorem for equivalence relation, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ 𝑅 = 𝑆    ⇒   ⊢ ( EqvRel 𝑅 ↔ EqvRel 𝑆)
Theorem	eqvreleqd 39378	Equality theorem for equivalence relation, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.)
⊢ (𝜑 → 𝑅 = 𝑆)    ⇒   ⊢ (𝜑 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
Theorem	eqvrelsym 39379	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)
Theorem	eqvrelsymb 39380	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 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴))
Theorem	eqvreltr 39381	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 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))
Theorem	eqvreltrd 39382	A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.)
⊢ (𝜑 → EqvRel 𝑅)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐵𝑅𝐶)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	eqvreltr4d 39383	A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Peter Mazsa, 2-Jun-2019.)
⊢ (𝜑 → EqvRel 𝑅)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    &   ⊢ (𝜑 → 𝐶𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐶)
Theorem	eqvrelref 39384	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)    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐴)
Theorem	eqvrelth 39385	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)    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
Theorem	eqvrelcl 39386	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)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ dom 𝑅)
Theorem	eqvrelthi 39387	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)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
Theorem	eqvreldisj 39388	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) = ∅))
Theorem	qsdisjALTV 39389	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)    &   ⊢ (𝜑 → 𝐵 ∈ (𝐴 / 𝑅))    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴 / 𝑅))    ⇒   ⊢ (𝜑 → (𝐵 = 𝐶 ∨ (𝐵 ∩ 𝐶) = ∅))
Theorem	eqvrelqsel 39390	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 𝑅 ∧ 𝐵 ∈ (𝐴 / 𝑅) ∧ 𝐶 ∈ 𝐵) → 𝐵 = [𝐶]𝑅)
Theorem	eqvrelcoss 39391	Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 4-Jul-2020.) (Revised by Peter Mazsa, 20-Dec-2021.)
⊢ ( EqvRel ≀ 𝑅 ↔ TrRel ≀ 𝑅)
Theorem	eqvrelcoss3 39392*	Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 28-Apr-2019.)
⊢ ( EqvRel ≀ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))
Theorem	eqvrelcoss2 39393	Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019.)
⊢ ( EqvRel ≀ 𝑅 ↔ ≀ ≀ 𝑅 ⊆ ≀ 𝑅)
Theorem	eqvrelcoss4 39394*	Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019.) (Revised by Peter Mazsa, 30-Sep-2021.)
⊢ ( EqvRel ≀ 𝑅 ↔ ∀𝑥∀𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅))
Theorem	dfcoeleqvrels 39395	Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 39393, eqvrelcoss3 39392 and eqvrelcoss4 39394 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }
Theorem	dfcoeleqvrel 39396	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 39393, eqvrelcoss3 39392 and eqvrelcoss4 39394 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)
⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)
21.26.15  Redundancy
Definition	df-redunds 39397*	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 39400) is equivalent to satisfying the redundancy predicate (df-redund 39398). (Contributed by Peter Mazsa, 23-Oct-2022.)
⊢ Redunds = ◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧))}
Definition	df-redund 39398	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 39400) is equivalent to satisfying the redundancy predicate. (Contributed by Peter Mazsa, 23-Oct-2022.)
⊢ (𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)))
Definition	df-redundp 39399	Define the redundancy operator for propositions, cf. df-redund 39398. (Contributed by Peter Mazsa, 23-Oct-2022.)
⊢ ( redund (𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒))))
Theorem	brredunds 39400	Binary relation on the class of all redundant sets. (Contributed by Peter Mazsa, 25-Oct-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))))