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Theorem List for Metamath Proof Explorer - 35801-35900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremidsymrel 35801 The identity relation is symmetric. (Contributed by AV, 19-Jun-2022.)
SymRel I
 
Theoremepnsymrel 35802 The membership (epsilon) relation is not symmetric. (Contributed by AV, 18-Jun-2022.)
¬ SymRel E
 
20.22.10  Reflexivity and symmetry
 
Theoremsymrefref2 35803 Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 35804. (Contributed by Peter Mazsa, 19-Jul-2018.)
(𝑅𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅))
 
Theoremsymrefref3 35804* Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref2 35803. (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 35805 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 35827) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 35757, cf. the comment of dfrefrels2 35757. (Contributed by Peter Mazsa, 20-Jul-2019.)
( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟𝑟𝑟)}
 
Theoremrefsymrels3 35806* 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 35828) can use the 𝑥 ∈ dom 𝑟𝑥𝑟𝑥 version for their reflexive part, not just the 𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥 = 𝑦𝑥𝑟𝑦) version of dfrefrels3 35758, cf. the comment of dfrefrel3 35760. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.)
( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥𝑦(𝑥𝑟𝑦𝑦𝑟𝑥))}
 
Theoremrefsymrel2 35807 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 35759, cf. the comment of dfrefrels2 35757. (Contributed by Peter Mazsa, 23-Aug-2021.)
(( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ Rel 𝑅))
 
Theoremrefsymrel3 35808* 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 35760, cf. the comment of dfrefrel3 35760. (Contributed by Peter Mazsa, 23-Aug-2021.)
(( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ Rel 𝑅))
 
Theoremelrefsymrels2 35809 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 35827) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrels2 35757, cf. the comment of dfrefrels2 35757. (Contributed by Peter Mazsa, 22-Jul-2019.)
(𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅𝑅𝑅) ∧ 𝑅 ∈ Rels ))
 
Theoremelrefsymrels3 35810* 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 35828) can use the 𝑥 ∈ dom 𝑅𝑥𝑅𝑥 version for their reflexive part, not just the 𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦𝑥𝑅𝑦) version of dfrefrels3 35758, cf. the comment of dfrefrel3 35760. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.)
(𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)) ∧ 𝑅 ∈ Rels ))
 
Theoremelrefsymrelsrel 35811 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 35812 Define the class of all transitive sets (versus the transitive class defined in df-tr 5176). It is used only by df-trrels 35813.

Note the similarity of the definitions of df-refs 35754, df-syms 35782 and df-trs 35812. (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 35813 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 35821. Alternate definitions are dftrrels2 35815 and dftrrels3 35816.

This definition is similar to the definitions of the classes of reflexive (df-refrels 35755) and symmetric (df-symrels 35783) relations. (Contributed by Peter Mazsa, 7-Jul-2019.)

TrRels = ( Trs ∩ Rels )
 
Definitiondf-trrel 35814 Define the transitive relation predicate. (Read: 𝑅 is a transitive relation.) For sets, being an element of the class of transitive relations (df-trrels 35813) is equivalent to satisfying the transitive relation predicate, see eltrrelsrel 35821. Alternate definitions are dftrrel2 35817 and dftrrel3 35818. (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 35815 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 5567 (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 35663 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 35816* Alternate definition of the class of transitive relations. (Contributed by Peter Mazsa, 22-Jul-2021.)
TrRels = {𝑟 ∈ Rels ∣ ∀𝑥𝑦𝑧((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)}
 
Theoremdftrrel2 35817 Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
( TrRel 𝑅 ↔ ((𝑅𝑅) ⊆ 𝑅 ∧ Rel 𝑅))
 
Theoremdftrrel3 35818* Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.)
( TrRel 𝑅 ↔ (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ Rel 𝑅))
 
Theoremeltrrels2 35819 Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ TrRels ↔ ((𝑅𝑅) ⊆ 𝑅𝑅 ∈ Rels ))
 
Theoremeltrrels3 35820* Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.)
(𝑅 ∈ TrRels ↔ (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ 𝑅 ∈ Rels ))
 
Theoremeltrrelsrel 35821 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 35822 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 35823 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 35833. Alternate definitions are dfeqvrels2 35827 and dfeqvrels3 35828. (Contributed by Peter Mazsa, 7-Nov-2018.)
EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels )
 
Definitiondf-eqvrel 35824 Define the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.) For sets, being an element of the class of equivalence relations (df-eqvrels 35823) is equivalent to satisfying the equivalence relation predicate, see eleqvrelsrel 35833. Alternate definitions are dfeqvrel2 35829 and dfeqvrel3 35830. (Contributed by Peter Mazsa, 17-Apr-2019.)
( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
 
Definitiondf-coeleqvrels 35825 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 35835. Alternate definition is dfcoeleqvrels 35860. (Contributed by Peter Mazsa, 28-Nov-2022.)
CoElEqvRels = {𝑎 ∣ ≀ ( E ↾ 𝑎) ∈ EqvRels }
 
Definitiondf-coeleqvrel 35826 Define the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.) Alternate definition is dfcoeleqvrel 35861. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 35835. (Contributed by Peter Mazsa, 11-Dec-2021.)
( CoElEqvRel 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
 
Theoremdfeqvrels2 35827 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 35828* 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 35829 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 35830* 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 35831 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 35832* 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 35833 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 35834 Elementhood in the coelement equivalence relations class. (Contributed by Peter Mazsa, 24-Jul-2023.)
(𝐴𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ ( E ↾ 𝐴) ∈ EqvRels ))
 
Theoremelcoeleqvrelsrel 35835 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 35836 An equivalence relation is a relation. (Contributed by Peter Mazsa, 2-Jun-2019.)
( EqvRel 𝑅 → Rel 𝑅)
 
Theoremeqvrelrefrel 35837 An equivalence relation is reflexive. (Contributed by Peter Mazsa, 29-Dec-2021.)
( EqvRel 𝑅 → RefRel 𝑅)
 
Theoremeqvrelsymrel 35838 An equivalence relation is symmetric. (Contributed by Peter Mazsa, 29-Dec-2021.)
( EqvRel 𝑅 → SymRel 𝑅)
 
Theoremeqvreltrrel 35839 An equivalence relation is transitive. (Contributed by Peter Mazsa, 29-Dec-2021.)
( EqvRel 𝑅 → TrRel 𝑅)
 
Theoremeqvrelim 35840 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 35841 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 35842 Equality theorem for equivalence relation, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.)
𝑅 = 𝑆       ( EqvRel 𝑅 ↔ EqvRel 𝑆)
 
Theoremeqvreleqd 35843 Equality theorem for equivalence relation, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.)
(𝜑𝑅 = 𝑆)       (𝜑 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
 
Theoremeqvrelsym 35844 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 35845 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 35846 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 35847 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 35848 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 35849 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 35850 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 35851 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 35852 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 35853 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 35854 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 35855 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 35856 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 35857* Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 28-Apr-2019.)
( EqvRel ≀ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
 
Theoremeqvrelcoss2 35858 Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019.)
( EqvRel ≀ 𝑅 ↔ ≀ ≀ 𝑅 ⊆ ≀ 𝑅)
 
Theoremeqvrelcoss4 35859* 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 35860 Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 35858, eqvrelcoss3 35857 and eqvrelcoss4 35859 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)
CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }
 
Theoremdfcoeleqvrel 35861 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 35858, eqvrelcoss3 35857 and eqvrelcoss4 35859 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)
( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)
 
20.22.13  Redundancy
 
Definitiondf-redunds 35862* 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 35865) is equivalent to satisfying the redundancy predicate (df-redund 35863). (Contributed by Peter Mazsa, 23-Oct-2022.)
Redunds = {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ (𝑥𝑦 ∧ (𝑥𝑧) = (𝑦𝑧))}
 
Definitiondf-redund 35863 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 35865) is equivalent to satisfying the redundancy predicate. (Contributed by Peter Mazsa, 23-Oct-2022.)
(𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ (𝐴𝐵 ∧ (𝐴𝐶) = (𝐵𝐶)))
 
Definitiondf-redundp 35864 Define the redundancy operator for propositions, cf. df-redund 35863. (Contributed by Peter Mazsa, 23-Oct-2022.)
( redund (𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ ((𝜑𝜒) ↔ (𝜓𝜒))))
 
Theorembrredunds 35865 Binary relation on the class of all redundant sets. (Contributed by Peter Mazsa, 25-Oct-2022.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ (𝐴𝐵 ∧ (𝐴𝐶) = (𝐵𝐶))))
 
Theorembrredundsredund 35866 For sets, binary relation on the class of all redundant sets (brredunds 35865) is equivalent to satisfying the redundancy predicate (df-redund 35863). (Contributed by Peter Mazsa, 25-Oct-2022.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝐴 Redunds ⟨𝐵, 𝐶⟩ ↔ 𝐴 Redund ⟨𝐵, 𝐶⟩))
 
Theoremredundss3 35867 Implication of redundancy predicate. (Contributed by Peter Mazsa, 26-Oct-2022.)
𝐷𝐶       (𝐴 Redund ⟨𝐵, 𝐶⟩ → 𝐴 Redund ⟨𝐵, 𝐷⟩)
 
Theoremredundeq1 35868 Equivalence of redundancy predicates. (Contributed by Peter Mazsa, 26-Oct-2022.)
𝐴 = 𝐷       (𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ 𝐷 Redund ⟨𝐵, 𝐶⟩)
 
Theoremredundpim3 35869 Implication of redundancy of proposition. (Contributed by Peter Mazsa, 26-Oct-2022.)
(𝜃𝜒)       ( redund (𝜑, 𝜓, 𝜒) → redund (𝜑, 𝜓, 𝜃))
 
Theoremredundpbi1 35870 Equivalence of redundancy of propositions. (Contributed by Peter Mazsa, 25-Oct-2022.)
(𝜑𝜃)       ( redund (𝜑, 𝜓, 𝜒) ↔ redund (𝜃, 𝜓, 𝜒))
 
Theoremrefrelsredund4 35871 The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 35757) if the relations are symmetric as well. (Contributed by Peter Mazsa, 26-Oct-2022.)
{𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , ( RefRels ∩ SymRels )⟩
 
Theoremrefrelsredund2 35872 The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 35757) in the class of equivalence relations. (Contributed by Peter Mazsa, 26-Oct-2022.)
{𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund ⟨ RefRels , EqvRels ⟩
 
Theoremrefrelsredund3 35873* The naive version of the class of reflexive relations {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑥𝑟𝑥} is redundant with respect to the class of reflexive relations (see dfrefrels3 35758) in the class of equivalence relations. (Contributed by Peter Mazsa, 26-Oct-2022.)
{𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} Redund ⟨ RefRels , EqvRels ⟩
 
Theoremrefrelredund4 35874 The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 35759) if the relation is symmetric as well. (Contributed by Peter Mazsa, 26-Oct-2022.)
redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅))
 
Theoremrefrelredund2 35875 The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 35759) in equivalence relation. (Contributed by Peter Mazsa, 25-Oct-2022.)
redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)
 
Theoremrefrelredund3 35876* The naive version of the definition of reflexive relation (∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 ∧ Rel 𝑅) is redundant with respect to reflexive relation (see dfrefrel3 35760) in equivalence relation. (Contributed by Peter Mazsa, 25-Oct-2022.)
redund ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)
 
20.22.14  Domain quotients
 
Definitiondf-dmqss 35877* Define the class of domain quotients. Domain quotients are pairs of sets, typically a relation and a set, where the quotient (see df-qs 8298) of the relation on its domain is equal to the set. See comments of df-ers 35901 for the motivation for this definition. (Contributed by Peter Mazsa, 16-Apr-2019.)
DomainQss = {⟨𝑥, 𝑦⟩ ∣ (dom 𝑥 / 𝑥) = 𝑦}
 
Definitiondf-dmqs 35878 Define the domain quotient predicate. (Read: the domain quotient of 𝑅 is 𝐴.) If 𝐴 and 𝑅 are sets, the domain quotient binary relation and the domain quotient predicate are the same, see brdmqssqs 35886. (Contributed by Peter Mazsa, 9-Aug-2021.)
(𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
 
Theoremdmqseq 35879 Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.)
(𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
 
Theoremdmqseqi 35880 Equality theorem for domain quotient, inference version. (Contributed by Peter Mazsa, 26-Sep-2021.)
𝑅 = 𝑆       (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆)
 
Theoremdmqseqd 35881 Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.)
(𝜑𝑅 = 𝑆)       (𝜑 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆))
 
Theoremdmqseqeq1 35882 Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.)
(𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))
 
Theoremdmqseqeq1i 35883 Equality theorem for domain quotient, inference version. (Contributed by Peter Mazsa, 26-Sep-2021.)
𝑅 = 𝑆       ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴)
 
Theoremdmqseqeq1d 35884 Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 26-Sep-2021.)
(𝜑𝑅 = 𝑆)       (𝜑 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))
 
Theorembrdmqss 35885 The domain quotient binary relation. (Contributed by Peter Mazsa, 17-Apr-2019.)
((𝐴𝑉𝑅𝑊) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴))
 
Theorembrdmqssqs 35886 If 𝐴 and 𝑅 are sets, the domain quotient binary relation and the domain quotient predicate are the same. (Contributed by Peter Mazsa, 14-Aug-2021.)
((𝐴𝑉𝑅𝑊) → (𝑅 DomainQss 𝐴𝑅 DomainQs 𝐴))
 
Theoremn0eldmqs 35887 The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 2-Mar-2018.)
¬ ∅ ∈ (dom 𝑅 / 𝑅)
 
Theoremn0eldmqseq 35888 The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 3-Nov-2018.)
((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴)
 
Theoremn0el3 35889 Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 27-May-2021.)
(¬ ∅ ∈ 𝐴 ↔ (dom ( E ↾ 𝐴) / ( E ↾ 𝐴)) = 𝐴)
 
Theoremcnvepresdmqss 35890 The domain quotient binary relation of the restricted converse epsilon relation is equivalent to the negated elementhood of the empty set in the restriction. (Contributed by Peter Mazsa, 14-Aug-2021.)
(𝐴𝑉 → (( E ↾ 𝐴) DomainQss 𝐴 ↔ ¬ ∅ ∈ 𝐴))
 
Theoremcnvepresdmqs 35891 The domain quotient predicate for the restricted converse epsilon relation is equivalent to the negated elementhood of the empty set in the restriction. (Contributed by Peter Mazsa, 14-Aug-2021.)
(( E ↾ 𝐴) DomainQs 𝐴 ↔ ¬ ∅ ∈ 𝐴)
 
Theoremunidmqs 35892 The range of a relation is equal to the union of the domain quotient. (Contributed by Peter Mazsa, 13-Oct-2018.)
(𝑅𝑉 → (Rel 𝑅 (dom 𝑅 / 𝑅) = ran 𝑅))
 
Theoremunidmqseq 35893 The union of the domain quotient of a relation is equal to the class 𝐴 if and only if the range is equal to it as well. (Contributed by Peter Mazsa, 21-Apr-2019.) (Revised by Peter Mazsa, 28-Dec-2021.)
(𝑅𝑉 → (Rel 𝑅 → ( (dom 𝑅 / 𝑅) = 𝐴 ↔ ran 𝑅 = 𝐴)))
 
Theoremdmqseqim 35894 If the domain quotient of a relation is equal to the class 𝐴, then the range of the relation is the union of the class. (Contributed by Peter Mazsa, 29-Dec-2021.)
(𝑅𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = 𝐴)))
 
Theoremdmqseqim2 35895 Lemma for erim2 35915. (Contributed by Peter Mazsa, 29-Dec-2021.)
(𝑅𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → (𝐵 ∈ ran 𝑅𝐵 𝐴))))
 
Theoremreleldmqs 35896* Elementhood in the domain quotient of a relation. (Contributed by Peter Mazsa, 24-Apr-2021.)
(𝐴𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅)))
 
Theoremeldmqs1cossres 35897* Elementhood in the domain quotient of the class of cosets by a restriction. (Contributed by Peter Mazsa, 4-May-2019.)
(𝐵𝑉 → (𝐵 ∈ (dom ≀ (𝑅𝐴) / ≀ (𝑅𝐴)) ↔ ∃𝑢𝐴𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅𝐴)))
 
Theoremreleldmqscoss 35898* Elementhood in the domain quotient of the class of cosets by a relation. (Contributed by Peter Mazsa, 23-Apr-2021.)
(𝐴𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom ≀ 𝑅 /𝑅) ↔ ∃𝑢 ∈ dom 𝑅𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅)))
 
Theoremdmqscoelseq 35899 Two ways to express the equality of the domain quotient of the coelements on the class 𝐴 with the class 𝐴. (Contributed by Peter Mazsa, 26-Sep-2021.)
((dom ∼ 𝐴 /𝐴) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)
 
Theoremdmqs1cosscnvepreseq 35900 Two ways to express the equality of the domain quotient of the coelements on the class 𝐴 with the class 𝐴. (Contributed by Peter Mazsa, 26-Sep-2021.)
((dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)
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