Home | Metamath
Proof Explorer Theorem List (p. 358 of 449) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-28623) |
Hilbert Space Explorer
(28624-30146) |
Users' Mathboxes
(30147-44804) |
Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-eqvrels 35701 | 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 35711. Alternate definitions are dfeqvrels2 35705 and dfeqvrels3 35706. (Contributed by Peter Mazsa, 7-Nov-2018.) |
⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels ) | ||
Definition | df-eqvrel 35702 | Define the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.) For sets, being an element of the class of equivalence relations (df-eqvrels 35701) is equivalent to satisfying the equivalence relation predicate, see eleqvrelsrel 35711. Alternate definitions are dfeqvrel2 35707 and dfeqvrel3 35708. (Contributed by Peter Mazsa, 17-Apr-2019.) |
⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅)) | ||
Definition | df-coeleqvrels 35703 | 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 35713. Alternate definition is dfcoeleqvrels 35738. (Contributed by Peter Mazsa, 28-Nov-2022.) |
⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } | ||
Definition | df-coeleqvrel 35704 | Define the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.) Alternate definition is dfcoeleqvrel 35739. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 35713. (Contributed by Peter Mazsa, 11-Dec-2021.) |
⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) | ||
Theorem | dfeqvrels2 35705 | 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 35706* | 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 35707 | 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 35708* | 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 35709 | 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 35710* | 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 35711 | 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 35712 | Elementhood in the coelement equivalence relations class. (Contributed by Peter Mazsa, 24-Jul-2023.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels )) | ||
Theorem | elcoeleqvrelsrel 35713 | 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 35714 | An equivalence relation is a relation. (Contributed by Peter Mazsa, 2-Jun-2019.) |
⊢ ( EqvRel 𝑅 → Rel 𝑅) | ||
Theorem | eqvrelrefrel 35715 | An equivalence relation is reflexive. (Contributed by Peter Mazsa, 29-Dec-2021.) |
⊢ ( EqvRel 𝑅 → RefRel 𝑅) | ||
Theorem | eqvrelsymrel 35716 | An equivalence relation is symmetric. (Contributed by Peter Mazsa, 29-Dec-2021.) |
⊢ ( EqvRel 𝑅 → SymRel 𝑅) | ||
Theorem | eqvreltrrel 35717 | An equivalence relation is transitive. (Contributed by Peter Mazsa, 29-Dec-2021.) |
⊢ ( EqvRel 𝑅 → TrRel 𝑅) | ||
Theorem | eqvrelim 35718 | 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 35719 | 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 35720 | Equality theorem for equivalence relation, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ 𝑅 = 𝑆 ⇒ ⊢ ( EqvRel 𝑅 ↔ EqvRel 𝑆) | ||
Theorem | eqvreleqd 35721 | Equality theorem for equivalence relation, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ (𝜑 → 𝑅 = 𝑆) ⇒ ⊢ (𝜑 → ( EqvRel 𝑅 ↔ EqvRel 𝑆)) | ||
Theorem | eqvrelsym 35722 | 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 35723 | 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 35724 | 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 35725 | 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 35726 | 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 35727 | 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 35728 | 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 35729 | 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 35730 | 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 35731 | 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 35732 | 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 35733 | 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 35734 | 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 35735* | 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 35736 | Two ways to express equivalent cosets. (Contributed by Peter Mazsa, 3-May-2019.) |
⊢ ( EqvRel ≀ 𝑅 ↔ ≀ ≀ 𝑅 ⊆ ≀ 𝑅) | ||
Theorem | eqvrelcoss4 35737* | 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 35738 | Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 35736, eqvrelcoss3 35735 and eqvrelcoss4 35737 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.) |
⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } | ||
Theorem | dfcoeleqvrel 35739 | 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 35736, eqvrelcoss3 35735 and eqvrelcoss4 35737 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.) |
⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴) | ||
Definition | df-redunds 35740* | 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 35743) is equivalent to satisfying the redundancy predicate (df-redund 35741). (Contributed by Peter Mazsa, 23-Oct-2022.) |
⊢ Redunds = ◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ (𝑥 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑧) = (𝑦 ∩ 𝑧))} | ||
Definition | df-redund 35741 | 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 35743) 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 35742 | Define the redundancy operator for propositions, cf. df-redund 35741. (Contributed by Peter Mazsa, 23-Oct-2022.) |
⊢ ( redund (𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒)))) | ||
Theorem | brredunds 35743 | 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)))) | ||
Theorem | brredundsredund 35744 | For sets, binary relation on the class of all redundant sets (brredunds 35743) is equivalent to satisfying the redundancy predicate (df-redund 35741). (Contributed by Peter Mazsa, 25-Oct-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 Redunds 〈𝐵, 𝐶〉 ↔ 𝐴 Redund 〈𝐵, 𝐶〉)) | ||
Theorem | redundss3 35745 | Implication of redundancy predicate. (Contributed by Peter Mazsa, 26-Oct-2022.) |
⊢ 𝐷 ⊆ 𝐶 ⇒ ⊢ (𝐴 Redund 〈𝐵, 𝐶〉 → 𝐴 Redund 〈𝐵, 𝐷〉) | ||
Theorem | redundeq1 35746 | Equivalence of redundancy predicates. (Contributed by Peter Mazsa, 26-Oct-2022.) |
⊢ 𝐴 = 𝐷 ⇒ ⊢ (𝐴 Redund 〈𝐵, 𝐶〉 ↔ 𝐷 Redund 〈𝐵, 𝐶〉) | ||
Theorem | redundpim3 35747 | Implication of redundancy of proposition. (Contributed by Peter Mazsa, 26-Oct-2022.) |
⊢ (𝜃 → 𝜒) ⇒ ⊢ ( redund (𝜑, 𝜓, 𝜒) → redund (𝜑, 𝜓, 𝜃)) | ||
Theorem | redundpbi1 35748 | Equivalence of redundancy of propositions. (Contributed by Peter Mazsa, 25-Oct-2022.) |
⊢ (𝜑 ↔ 𝜃) ⇒ ⊢ ( redund (𝜑, 𝜓, 𝜒) ↔ redund (𝜃, 𝜓, 𝜒)) | ||
Theorem | refrelsredund4 35749 | The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 35635) if the relations are symmetric as well. (Contributed by Peter Mazsa, 26-Oct-2022.) |
⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund 〈 RefRels , ( RefRels ∩ SymRels )〉 | ||
Theorem | refrelsredund2 35750 | The naive version of the class of reflexive relations is redundant with respect to the class of reflexive relations (see dfrefrels2 35635) in the class of equivalence relations. (Contributed by Peter Mazsa, 26-Oct-2022.) |
⊢ {𝑟 ∈ Rels ∣ ( I ↾ dom 𝑟) ⊆ 𝑟} Redund 〈 RefRels , EqvRels 〉 | ||
Theorem | refrelsredund3 35751* | 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 35636) in the class of equivalence relations. (Contributed by Peter Mazsa, 26-Oct-2022.) |
⊢ {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥} Redund 〈 RefRels , EqvRels 〉 | ||
Theorem | refrelredund4 35752 | The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 35637) 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)) | ||
Theorem | refrelredund2 35753 | The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 35637) in equivalence relation. (Contributed by Peter Mazsa, 25-Oct-2022.) |
⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅) | ||
Theorem | refrelredund3 35754* | The naive version of the definition of reflexive relation (∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 ∧ Rel 𝑅) is redundant with respect to reflexive relation (see dfrefrel3 35638) in equivalence relation. (Contributed by Peter Mazsa, 25-Oct-2022.) |
⊢ redund ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅) | ||
Definition | df-dmqss 35755* | Define the class of domain quotients. Domain quotients are pairs of sets, typically a relation and a set, where the quotient (see df-qs 8285) of the relation on its domain is equal to the set. See comments of df-ers 35779 for the motivation for this definition. (Contributed by Peter Mazsa, 16-Apr-2019.) |
⊢ DomainQss = {〈𝑥, 𝑦〉 ∣ (dom 𝑥 / 𝑥) = 𝑦} | ||
Definition | df-dmqs 35756 | 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 35764. (Contributed by Peter Mazsa, 9-Aug-2021.) |
⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴) | ||
Theorem | dmqseq 35757 | Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.) |
⊢ (𝑅 = 𝑆 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆)) | ||
Theorem | dmqseqi 35758 | Equality theorem for domain quotient, inference version. (Contributed by Peter Mazsa, 26-Sep-2021.) |
⊢ 𝑅 = 𝑆 ⇒ ⊢ (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆) | ||
Theorem | dmqseqd 35759 | Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 23-Apr-2021.) |
⊢ (𝜑 → 𝑅 = 𝑆) ⇒ ⊢ (𝜑 → (dom 𝑅 / 𝑅) = (dom 𝑆 / 𝑆)) | ||
Theorem | dmqseqeq1 35760 | Equality theorem for domain quotient. (Contributed by Peter Mazsa, 17-Apr-2019.) |
⊢ (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴)) | ||
Theorem | dmqseqeq1i 35761 | Equality theorem for domain quotient, inference version. (Contributed by Peter Mazsa, 26-Sep-2021.) |
⊢ 𝑅 = 𝑆 ⇒ ⊢ ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴) | ||
Theorem | dmqseqeq1d 35762 | Equality theorem for domain quotient set, deduction version. (Contributed by Peter Mazsa, 26-Sep-2021.) |
⊢ (𝜑 → 𝑅 = 𝑆) ⇒ ⊢ (𝜑 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴)) | ||
Theorem | brdmqss 35763 | The domain quotient binary relation. (Contributed by Peter Mazsa, 17-Apr-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 DomainQss 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)) | ||
Theorem | brdmqssqs 35764 | 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 DomainQss 𝐴 ↔ 𝑅 DomainQs 𝐴)) | ||
Theorem | n0eldmqs 35765 | The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 2-Mar-2018.) |
⊢ ¬ ∅ ∈ (dom 𝑅 / 𝑅) | ||
Theorem | n0eldmqseq 35766 | The empty set is not an element of a domain quotient. (Contributed by Peter Mazsa, 3-Nov-2018.) |
⊢ ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴) | ||
Theorem | n0el3 35767 | 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) | ||
Theorem | cnvepresdmqss 35768 | 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.) |
⊢ (𝐴 ∈ 𝑉 → ((◡ E ↾ 𝐴) DomainQss 𝐴 ↔ ¬ ∅ ∈ 𝐴)) | ||
Theorem | cnvepresdmqs 35769 | 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) | ||
Theorem | unidmqs 35770 | The range of a relation is equal to the union of the domain quotient. (Contributed by Peter Mazsa, 13-Oct-2018.) |
⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ∪ (dom 𝑅 / 𝑅) = ran 𝑅)) | ||
Theorem | unidmqseq 35771 | 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.) |
⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → (∪ (dom 𝑅 / 𝑅) = 𝐴 ↔ ran 𝑅 = 𝐴))) | ||
Theorem | dmqseqim 35772 | 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.) |
⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = ∪ 𝐴))) | ||
Theorem | dmqseqim2 35773 | Lemma for erim2 35793. (Contributed by Peter Mazsa, 29-Dec-2021.) |
⊢ (𝑅 ∈ 𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → (𝐵 ∈ ran 𝑅 ↔ 𝐵 ∈ ∪ 𝐴)))) | ||
Theorem | releldmqs 35774* | Elementhood in the domain quotient of a relation. (Contributed by Peter Mazsa, 24-Apr-2021.) |
⊢ (𝐴 ∈ 𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom 𝑅 / 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑢]𝑅))) | ||
Theorem | eldmqs1cossres 35775* | Elementhood in the domain quotient of the class of cosets by a restriction. (Contributed by Peter Mazsa, 4-May-2019.) |
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom ≀ (𝑅 ↾ 𝐴) / ≀ (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑥] ≀ (𝑅 ↾ 𝐴))) | ||
Theorem | releldmqscoss 35776* | Elementhood in the domain quotient of the class of cosets by a relation. (Contributed by Peter Mazsa, 23-Apr-2021.) |
⊢ (𝐴 ∈ 𝑉 → (Rel 𝑅 → (𝐴 ∈ (dom ≀ 𝑅 / ≀ 𝑅) ↔ ∃𝑢 ∈ dom 𝑅∃𝑥 ∈ [ 𝑢]𝑅𝐴 = [𝑥] ≀ 𝑅))) | ||
Theorem | dmqscoelseq 35777 | 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) | ||
Theorem | dmqs1cosscnvepreseq 35778 | 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) | ||
Definition | df-ers 35779 |
Define the class of equivalence relations on domain quotients (or: domain
quotients restricted to equivalence relations).
The present definition of equivalence relation in set.mm df-er 8279 "is not standard", "somewhat cryptic", has no costant 0-ary class and does not follow the traditional transparent reflexive-symmetric-transitive relation way of definition of equivalence. The definitions df-eqvrels 35701, dfeqvrels2 35705, dfeqvrels3 35706 and df-eqvrel 35702, dfeqvrel2 35707, dfeqvrel3 35708 are fully transparent in this regard. However, they lack the domain component (dom 𝑅 = 𝐴) of the present df-er 8279. While we acknowledge the need of a domain component, the present df-er 8279 definition does not utilize the results revealed by the new theorems in the Partition-Equivalence Theorem part below (like ~? pets and ~? pet ). From those theorems follows that the natural domain of equivalence relations is not 𝑅Domain𝐴 (i.e. dom 𝑅 = 𝐴 see brdomaing 33294), but 𝑅 DomainQss 𝐴 (i.e. (dom 𝑅 / 𝑅) = 𝐴, see brdmqss 35763), see erim 35794 vs. prter3 35900. While I'm sure we need both equivalence relation df-eqvrels 35701 and equivalence relation on domain quotient df-ers 35779, I'm not sure whether we need a third equivalence relation concept with the present dom 𝑅 = 𝐴 component as well: this needs further investigation. As a default I suppose that these two concepts df-eqvrels 35701 and df-ers 35779 are enough and named the predicate version of the one on domain quotient as the alternate version df-erALTV 35780 of the present df-er 8279. (Contributed by Peter Mazsa, 26-Jun-2021.) |
⊢ Ers = ( DomainQss ↾ EqvRels ) | ||
Definition | df-erALTV 35780 | Equivalence relation with natural domain predicate, see also the comment of df-ers 35779. Alternate definition is dferALTV2 35784. Binary equivalence relation with natural domain and the equivalence relation with natural domain predicate are the same when 𝐴 and 𝑅 are sets, see brerser 35792. (Contributed by Peter Mazsa, 12-Aug-2021.) |
⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴)) | ||
Definition | df-members 35781 | Define the class of membership equivalence relations on their domain quotients. (Contributed by Peter Mazsa, 28-Nov-2022.) (Revised by Peter Mazsa, 24-Jul-2023.) |
⊢ MembErs = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎} | ||
Definition | df-member 35782 |
Define the membership equivalence relation on the class 𝐴 (or, the
restricted elementhood equivalence relation on its domain quotient
𝐴.) Alternate definitions are dfmember2 35789 and dfmember3 35790.
Later on, in an application of set theory I make a distinction between the default elementhood concept and a special membership concept: membership equivalence relation will be an integral part of that membership concept. (Contributed by Peter Mazsa, 26-Jun-2021.) (Revised by Peter Mazsa, 28-Nov-2022.) |
⊢ ( MembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴) | ||
Theorem | brers 35783 | Binary equivalence relation with natural domain, see the comment of df-ers 35779. (Contributed by Peter Mazsa, 23-Jul-2021.) |
⊢ (𝐴 ∈ 𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴))) | ||
Theorem | dferALTV2 35784 | Equivalence relation with natural domain predicate, see the comment of df-ers 35779. (Contributed by Peter Mazsa, 26-Jun-2021.) (Revised by Peter Mazsa, 30-Aug-2021.) |
⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) | ||
Theorem | erALTVeq1 35785 | Equality theorem for equivalence relation on domain quotient. (Contributed by Peter Mazsa, 25-Sep-2021.) |
⊢ (𝑅 = 𝑆 → (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴)) | ||
Theorem | erALTVeq1i 35786 | Equality theorem for equivalence relation on domain quotient, inference version. (Contributed by Peter Mazsa, 25-Sep-2021.) |
⊢ 𝑅 = 𝑆 ⇒ ⊢ (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴) | ||
Theorem | erALTVeq1d 35787 | Equality theorem for equivalence relation on domain quotient, deduction version. (Contributed by Peter Mazsa, 25-Sep-2021.) |
⊢ (𝜑 → 𝑅 = 𝑆) ⇒ ⊢ (𝜑 → (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴)) | ||
Theorem | dfmember 35788 | Alternate definition of the membership equivalence relation. (Contributed by Peter Mazsa, 28-Nov-2022.) |
⊢ ( MembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴) | ||
Theorem | dfmember2 35789 | Alternate definition of the membership equivalence relation. (Contributed by Peter Mazsa, 25-Sep-2021.) |
⊢ ( MembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴)) | ||
Theorem | dfmember3 35790 | Alternate definition of the membership equivalence relation. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, 17-Jul-2023.) |
⊢ ( MembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) | ||
Theorem | eqvreldmqs 35791 | Two ways to express membership equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, 17-Jul-2023.) |
⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) | ||
Theorem | brerser 35792 | Binary equivalence relation with natural domain and the equivalence relation with natural domain predicate are the same when 𝐴 and 𝑅 are sets. (Contributed by Peter Mazsa, 25-Aug-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 Ers 𝐴 ↔ 𝑅 ErALTV 𝐴)) | ||
Theorem | erim2 35793 | Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is prter3 35900 in a more convenient form , see also erim 35794). (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 29-Dec-2021.) |
⊢ (𝑅 ∈ 𝑉 → (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ∼ 𝐴 = 𝑅)) | ||
Theorem | erim 35794 | Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is the most convenient form of prter3 35900 and erim2 35793). (Contributed by Peter Mazsa, 7-Oct-2021.) (Revised by Peter Mazsa, 29-Dec-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ErALTV 𝐴 → ∼ 𝐴 = 𝑅)) | ||
Definition | df-funss 35795 | Define the class of all function sets (but not necessarily function relations, cf. df-funsALTV 35796). It is used only by df-funsALTV 35796. (Contributed by Peter Mazsa, 17-Jul-2021.) |
⊢ Funss = {𝑥 ∣ ≀ 𝑥 ∈ CnvRefRels } | ||
Definition | df-funsALTV 35796 | Define the function relations class, i.e., the class of functions. Alternate definitions are dffunsALTV 35798, ... , dffunsALTV5 35802. (Contributed by Peter Mazsa, 17-Jul-2021.) |
⊢ FunsALTV = ( Funss ∩ Rels ) | ||
Definition | df-funALTV 35797 |
Define the function relation predicate, i.e., the function predicate.
This definition of the function predicate (based on a more general,
converse reflexive, relation) and the original definition of function in
set.mm df-fun 6351, are always the same, that is
( FunALTV 𝐹 ↔ Fun 𝐹), see funALTVfun 35813.
The element of the class of functions and the function predicate are the same, that is (𝐹 ∈ FunsALTV ↔ FunALTV 𝐹) when 𝐹 is a set, see elfunsALTVfunALTV 35812. Alternate definitions are dffunALTV2 35803, ... , dffunALTV5 35806. (Contributed by Peter Mazsa, 17-Jul-2021.) |
⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹)) | ||
Theorem | dffunsALTV 35798 | Alternate definition of the class of functions. (Contributed by Peter Mazsa, 18-Jul-2021.) |
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ∈ CnvRefRels } | ||
Theorem | dffunsALTV2 35799 | Alternate definition of the class of functions. (Contributed by Peter Mazsa, 30-Aug-2021.) |
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ≀ 𝑓 ⊆ I } | ||
Theorem | dffunsALTV3 35800* | Alternate definition of the class of functions. For the 𝑋 axis and the 𝑌 axis you can convert the right side to {𝑓 ∈ Rels ∣ ∀ x1 ∀ y1 ∀ y2 (( x1 𝑓 y1 ∧ x1 𝑓 y2 ) → y1 = y2 )}. (Contributed by Peter Mazsa, 30-Aug-2021.) |
⊢ FunsALTV = {𝑓 ∈ Rels ∣ ∀𝑢∀𝑥∀𝑦((𝑢𝑓𝑥 ∧ 𝑢𝑓𝑦) → 𝑥 = 𝑦)} |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |