Home | Metamath
Proof Explorer Theorem List (p. 357 of 450) | < 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
(1-28697) |
Hilbert Space Explorer
(28698-30220) |
Users' Mathboxes
(30221-44913) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | brcnvrabga 35601* | The law of concretion for the converse of operation class abstraction. (Contributed by Peter Mazsa, 25-Oct-2022.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) & ⊢ 𝑅 = ◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑅〈𝐵, 𝐶〉 ↔ 𝜓)) | ||
Theorem | opideq 35602 | Equality conditions for ordered pairs 〈𝐴, 𝐴〉 and 〈𝐵, 𝐵〉. (Contributed by Peter Mazsa, 22-Jul-2019.) (Revised by Thierry Arnoux, 16-Feb-2022.) |
⊢ (𝐴 ∈ 𝑉 → (〈𝐴, 𝐴〉 = 〈𝐵, 𝐵〉 ↔ 𝐴 = 𝐵)) | ||
Theorem | iss2 35603 | A subclass of the identity relation is the intersection of identity relation with Cartesian product of the domain and range of the class. (Contributed by Peter Mazsa, 22-Jul-2019.) |
⊢ (𝐴 ⊆ I ↔ 𝐴 = ( I ∩ (dom 𝐴 × ran 𝐴))) | ||
Theorem | eldmcnv 35604* | Elementhood in a domain of a converse. (Contributed by Peter Mazsa, 25-May-2018.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) | ||
Theorem | dfrel5 35605 | Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 6-Nov-2018.) |
⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅) | ||
Theorem | dfrel6 35606 | Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 14-Mar-2019.) |
⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅) | ||
Theorem | cnvresrn 35607 | Converse restricted to range is converse. (Contributed by Peter Mazsa, 3-Sep-2021.) |
⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅 | ||
Theorem | ecin0 35608* | Two ways of saying that the coset of 𝐴 and the coset of 𝐵 have no elements in common. (Contributed by Peter Mazsa, 1-Dec-2018.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ↔ ∀𝑥(𝐴𝑅𝑥 → ¬ 𝐵𝑅𝑥))) | ||
Theorem | ecinn0 35609* | Two ways of saying that the coset of 𝐴 and the coset of 𝐵 have some elements in common. (Contributed by Peter Mazsa, 23-Jan-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) | ||
Theorem | ineleq 35610* | Equivalence of restricted universal quantifications. (Contributed by Peter Mazsa, 29-May-2018.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑧∀𝑦 ∈ 𝐵 ((𝑧 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷) → 𝑥 = 𝑦)) | ||
Theorem | inecmo 35611* | Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 29-May-2018.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (Rel 𝑅 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧)) | ||
Theorem | inecmo2 35612* | Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 29-May-2018.) (Revised by Peter Mazsa, 2-Sep-2021.) |
⊢ ((∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢 ∈ 𝐴 𝑢𝑅𝑥 ∧ Rel 𝑅)) | ||
Theorem | ineccnvmo 35613* | Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 2-Sep-2021.) |
⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦) | ||
Theorem | alrmomorn 35614 | Equivalence of an "at most one" and an "at most one" restricted to the range inside a universal quantification. (Contributed by Peter Mazsa, 3-Sep-2021.) |
⊢ (∀𝑥∃*𝑦 ∈ ran 𝑅 𝑥𝑅𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝑅𝑦) | ||
Theorem | alrmomodm 35615* | Equivalence of an "at most one" and an "at most one" restricted to the domain inside a universal quantification. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ (Rel 𝑅 → (∀𝑥∃*𝑢 ∈ dom 𝑅 𝑢𝑅𝑥 ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥)) | ||
Theorem | ineccnvmo2 35616* | Equivalence of a double universal quantification restricted to the range and an "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 4-Sep-2021.) |
⊢ (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ ran 𝐹(𝑥 = 𝑦 ∨ ([𝑥]◡𝐹 ∩ [𝑦]◡𝐹) = ∅) ↔ ∀𝑢∃*𝑥 𝑢𝐹𝑥) | ||
Theorem | inecmo3 35617* | Equivalence of a double universal quantification restricted to the domain and an "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ ((∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢 𝑢𝑅𝑥 ∧ Rel 𝑅)) | ||
Theorem | moantr 35618 | Sufficient condition for transitivity of conjunctions inside existential quantifiers. (Contributed by Peter Mazsa, 2-Oct-2018.) |
⊢ (∃*𝑥𝜓 → ((∃𝑥(𝜑 ∧ 𝜓) ∧ ∃𝑥(𝜓 ∧ 𝜒)) → ∃𝑥(𝜑 ∧ 𝜒))) | ||
Theorem | brabidgaw 35619* | The law of concretion for a binary relation. Special case of brabga 5423. Version of brabidga 35620 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by Peter Mazsa, 24-Nov-2018.) (Revised by Gino Giotto, 2-Apr-2024.) |
⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ (𝑥𝑅𝑦 ↔ 𝜑) | ||
Theorem | brabidga 35620 | The law of concretion for a binary relation. Special case of brabga 5423. Usage of this theorem is discouraged because it depends on ax-13 2390, see brabidgaw 35619 for a weaker version that does not require it. (Contributed by Peter Mazsa, 24-Nov-2018.) (New usage is discouraged.) |
⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ (𝑥𝑅𝑦 ↔ 𝜑) | ||
Theorem | inxp2 35621* | Intersection with a Cartesian product. (Contributed by Peter Mazsa, 18-Jul-2019.) |
⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)} | ||
Theorem | opabf 35622 | A class abstraction of a collection of ordered pairs with a negated wff is the empty set. (Contributed by Peter Mazsa, 21-Oct-2019.) (Proof shortened by Thierry Arnoux, 18-Feb-2022.) |
⊢ ¬ 𝜑 ⇒ ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = ∅ | ||
Theorem | ec0 35623 | The empty-coset of a class is the empty set. (Contributed by Peter Mazsa, 19-May-2019.) |
⊢ [𝐴]∅ = ∅ | ||
Theorem | 0qs 35624 | Quotient set with the empty set. (Contributed by Peter Mazsa, 14-Sep-2019.) |
⊢ (∅ / 𝑅) = ∅ | ||
Definition | df-xrn 35625 | Define the range Cartesian product of two classes. Definition from [Holmes] p. 40. Membership in this class is characterized by xrnss3v 35626 and brxrn 35628. This is Scott Fenton's df-txp 33317 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 33317. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ (𝐴 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) | ||
Theorem | xrnss3v 35626 | A range Cartesian product is a subset of the class of ordered triples. This is Scott Fenton's txpss3v 33341 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 33341. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V)) | ||
Theorem | xrnrel 35627 | A range Cartesian product is a relation. This is Scott Fenton's txprel 33342 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 33342. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ Rel (𝐴 ⋉ 𝐵) | ||
Theorem | brxrn 35628 | Characterize a ternary relation over a range Cartesian product. Together with xrnss3v 35626, this characterizes elementhood in a range cross. (Contributed by Peter Mazsa, 27-Jun-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴(𝑅 ⋉ 𝑆)〈𝐵, 𝐶〉 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐶))) | ||
Theorem | brxrn2 35629* | A characterization of the range Cartesian product. (Contributed by Peter Mazsa, 14-Oct-2020.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) | ||
Theorem | dfxrn2 35630* | Alternate definition of the range Cartesian product. (Contributed by Peter Mazsa, 20-Feb-2022.) |
⊢ (𝑅 ⋉ 𝑆) = ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} | ||
Theorem | xrneq1 35631 | Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.) |
⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)) | ||
Theorem | xrneq1i 35632 | Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶) | ||
Theorem | xrneq1d 35633 | Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)) | ||
Theorem | xrneq2 35634 | Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.) |
⊢ (𝐴 = 𝐵 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵)) | ||
Theorem | xrneq2i 35635 | Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵) | ||
Theorem | xrneq2d 35636 | Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 7-Sep-2021.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵)) | ||
Theorem | xrneq12 35637 | Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)) | ||
Theorem | xrneq12i 35638 | Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷) | ||
Theorem | xrneq12d 35639 | Equality theorem for the range Cartesian product, deduction form. (Contributed by Peter Mazsa, 18-Dec-2021.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)) | ||
Theorem | elecxrn 35640* | Elementhood in the (𝑅 ⋉ 𝑆)-coset of 𝐴. (Contributed by Peter Mazsa, 18-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.) |
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ ∃𝑥∃𝑦(𝐵 = 〈𝑥, 𝑦〉 ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦))) | ||
Theorem | ecxrn 35641* | The (𝑅 ⋉ 𝑆)-coset of 𝐴. (Contributed by Peter Mazsa, 18-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.) |
⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {〈𝑦, 𝑧〉 ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)}) | ||
Theorem | xrninxp 35642* | Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 7-Apr-2020.) |
⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = ◡{〈〈𝑦, 𝑧〉, 𝑢〉 ∣ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝑦, 𝑧〉))} | ||
Theorem | xrninxp2 35643* | Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 8-Apr-2020.) |
⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {〈𝑢, 𝑥〉 ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))} | ||
Theorem | xrninxpex 35644 | Sufficient condition for the intersection of a range Cartesian product with a Cartesian product to be a set. (Contributed by Peter Mazsa, 12-Apr-2020.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V) | ||
Theorem | inxpxrn 35645 | Two ways to express the intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 10-Apr-2020.) |
⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶))) = ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) | ||
Theorem | br1cnvxrn2 35646* | The converse of a binary relation over a range Cartesian product. (Contributed by Peter Mazsa, 11-Jul-2021.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴◡(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑦∃𝑧(𝐴 = 〈𝑦, 𝑧〉 ∧ 𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))) | ||
Theorem | elec1cnvxrn2 35647* | Elementhood in the converse range Cartesian product coset of 𝐴. (Contributed by Peter Mazsa, 11-Jul-2021.) |
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ [𝐴]◡(𝑅 ⋉ 𝑆) ↔ ∃𝑦∃𝑧(𝐴 = 〈𝑦, 𝑧〉 ∧ 𝐵𝑅𝑦 ∧ 𝐵𝑆𝑧))) | ||
Theorem | rnxrn 35648* | Range of the range Cartesian product of classes. (Contributed by Peter Mazsa, 1-Jun-2020.) |
⊢ ran (𝑅 ⋉ 𝑆) = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} | ||
Theorem | rnxrnres 35649* | Range of a range Cartesian product with a restricted relation. (Contributed by Peter Mazsa, 5-Dec-2021.) |
⊢ ran (𝑅 ⋉ (𝑆 ↾ 𝐴)) = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} | ||
Theorem | rnxrncnvepres 35650* | Range of a range Cartesian product with a restriction of the converse epsilon relation. (Contributed by Peter Mazsa, 6-Dec-2021.) |
⊢ ran (𝑅 ⋉ (◡ E ↾ 𝐴)) = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ 𝑢 ∧ 𝑢𝑅𝑥)} | ||
Theorem | rnxrnidres 35651* | Range of a range Cartesian product with a restriction of the identity relation. (Contributed by Peter Mazsa, 6-Dec-2021.) |
⊢ ran (𝑅 ⋉ ( I ↾ 𝐴)) = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢 = 𝑦 ∧ 𝑢𝑅𝑥)} | ||
Theorem | xrnres 35652 | Two ways to express restriction of range Cartesian product, see also xrnres2 35653, xrnres3 35654. (Contributed by Peter Mazsa, 5-Jun-2021.) |
⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ 𝑆) | ||
Theorem | xrnres2 35653 | Two ways to express restriction of range Cartesian product, see also xrnres 35652, xrnres3 35654. (Contributed by Peter Mazsa, 6-Sep-2021.) |
⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴)) | ||
Theorem | xrnres3 35654 | Two ways to express restriction of range Cartesian product, see also xrnres 35652, xrnres2 35653. (Contributed by Peter Mazsa, 28-Mar-2020.) |
⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) | ||
Theorem | xrnres4 35655 | Two ways to express restriction of range Cartesian product. (Contributed by Peter Mazsa, 29-Dec-2020.) |
⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (ran (𝑅 ↾ 𝐴) × ran (𝑆 ↾ 𝐴)))) | ||
Theorem | xrnresex 35656 | Sufficient condition for a restricted range Cartesian product to be a set. (Contributed by Peter Mazsa, 16-Dec-2020.) (Revised by Peter Mazsa, 7-Sep-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ (𝑆 ↾ 𝐴) ∈ 𝑋) → (𝑅 ⋉ (𝑆 ↾ 𝐴)) ∈ V) | ||
Theorem | xrnidresex 35657 | Sufficient condition for a range Cartesian product with restricted identity to be a set. (Contributed by Peter Mazsa, 31-Dec-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 ⋉ ( I ↾ 𝐴)) ∈ V) | ||
Theorem | xrncnvepresex 35658 | Sufficient condition for a range Cartesian product with restricted converse epsilon to be a set. (Contributed by Peter Mazsa, 16-Dec-2020.) (Revised by Peter Mazsa, 23-Sep-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V) | ||
Theorem | brin2 35659 | Binary relation on an intersection is a special case of binary relation on range Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆)〈𝐵, 𝐵〉)) | ||
Theorem | brin3 35660 | Binary relation on an intersection is a special case of binary relation on range Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021.) (Avoid depending on this detail.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆){{𝐵}})) | ||
Definition | df-coss 35661* |
Define the class of cosets by 𝑅: 𝑥 and 𝑦 are cosets by
𝑅 iff there exists a set 𝑢 such
that both 𝑢𝑅𝑥 and
𝑢𝑅𝑦 hold, i.e., both 𝑥 and
𝑦
are are elements of the 𝑅
-coset of 𝑢 (see dfcoss2 35663 and the comment of dfec2 8294). 𝑅 is
usually a relation.
This concept simplifies theorems relating partition and equivalence: the left side of these theorems relate to 𝑅, the right side relate to ≀ 𝑅 (see e.g. ~? pet ). Without the definition of ≀ 𝑅 we should have to relate the right side of these theorems to a composition of a converse (cf. dfcoss3 35664) or to the range of a range Cartesian product of classes (cf. dfcoss4 35665), which would make the theorems complicated and confusing. Alternate definition is dfcoss2 35663. Technically, we can define it via composition (dfcoss3 35664) or as the range of a range Cartesian product (dfcoss4 35665), but neither of these definitions reveal directly how the cosets by 𝑅 relate to each other. We define functions (df-funsALTV 35916, df-funALTV 35917) and disjoints (dfdisjs 35943, dfdisjs2 35944, df-disjALTV 35940, dfdisjALTV2 35949) with the help of it as well. (Contributed by Peter Mazsa, 9-Jan-2018.) |
⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | ||
Definition | df-coels 35662 | Define the class of coelements on the class 𝐴, see also the alternate definition dfcoels 35677. Possible definitions are the special cases of dfcoss3 35664 and dfcoss4 35665. (Contributed by Peter Mazsa, 20-Nov-2019.) |
⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | ||
Theorem | dfcoss2 35663* | Alternate definition of the class of cosets by 𝑅: 𝑥 and 𝑦 are cosets by 𝑅 iff there exists a set 𝑢 such that both 𝑥 and 𝑦 are are elements of the 𝑅-coset of 𝑢 (see also the comment of dfec2 8294). 𝑅 is usually a relation. (Contributed by Peter Mazsa, 16-Jan-2018.) |
⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)} | ||
Theorem | dfcoss3 35664 | Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 35661). (Contributed by Peter Mazsa, 27-Dec-2018.) |
⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅) | ||
Theorem | dfcoss4 35665 | Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 35661). (Contributed by Peter Mazsa, 12-Jul-2021.) |
⊢ ≀ 𝑅 = ran (𝑅 ⋉ 𝑅) | ||
Theorem | cossex 35666 | If 𝐴 is a set then the class of cosets by 𝐴 is a set. (Contributed by Peter Mazsa, 4-Jan-2019.) |
⊢ (𝐴 ∈ 𝑉 → ≀ 𝐴 ∈ V) | ||
Theorem | cosscnvex 35667 | If 𝐴 is a set then the class of cosets by the converse of 𝐴 is a set. (Contributed by Peter Mazsa, 18-Oct-2019.) |
⊢ (𝐴 ∈ 𝑉 → ≀ ◡𝐴 ∈ V) | ||
Theorem | 1cosscnvepresex 35668 | Sufficient condition for a restricted converse epsilon coset to be a set. (Contributed by Peter Mazsa, 24-Sep-2021.) |
⊢ (𝐴 ∈ 𝑉 → ≀ (◡ E ↾ 𝐴) ∈ V) | ||
Theorem | 1cossxrncnvepresex 35669 | Sufficient condition for a restricted converse epsilon range Cartesian product to be a set. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V) | ||
Theorem | relcoss 35670 | Cosets by 𝑅 is a relation. (Contributed by Peter Mazsa, 27-Dec-2018.) |
⊢ Rel ≀ 𝑅 | ||
Theorem | relcoels 35671 | Coelements on 𝐴 is a relation. (Contributed by Peter Mazsa, 5-Oct-2021.) |
⊢ Rel ∼ 𝐴 | ||
Theorem | cossss 35672 | Subclass theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 11-Nov-2019.) |
⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵) | ||
Theorem | cosseq 35673 | Equality theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 9-Jan-2018.) |
⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵) | ||
Theorem | cosseqi 35674 | Equality theorem for the classes of cosets by 𝐴 and 𝐵, inference form. (Contributed by Peter Mazsa, 9-Jan-2018.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ≀ 𝐴 = ≀ 𝐵 | ||
Theorem | cosseqd 35675 | Equality theorem for the classes of cosets by 𝐴 and 𝐵, deduction form. (Contributed by Peter Mazsa, 4-Nov-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ≀ 𝐴 = ≀ 𝐵) | ||
Theorem | 1cossres 35676* | The class of cosets by a restriction. (Contributed by Peter Mazsa, 20-Apr-2019.) |
⊢ ≀ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | ||
Theorem | dfcoels 35677* | Alternate definition of the class of coelements on the class 𝐴. (Contributed by Peter Mazsa, 20-Apr-2019.) |
⊢ ∼ 𝐴 = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢)} | ||
Theorem | brcoss 35678* | 𝐴 and 𝐵 are cosets by 𝑅: a binary relation. (Contributed by Peter Mazsa, 27-Dec-2018.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵))) | ||
Theorem | brcoss2 35679* | Alternate form of the 𝐴 and 𝐵 are cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 26-Mar-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝐴 ∈ [𝑢]𝑅 ∧ 𝐵 ∈ [𝑢]𝑅))) | ||
Theorem | brcoss3 35680 | Alternate form of the 𝐴 and 𝐵 are cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 26-Mar-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ([𝐴]◡𝑅 ∩ [𝐵]◡𝑅) ≠ ∅)) | ||
Theorem | brcosscnvcoss 35681 | For sets, the 𝐴 and 𝐵 cosets by 𝑅 binary relation and the 𝐵 and 𝐴 cosets by 𝑅 binary relation are the same. (Contributed by Peter Mazsa, 27-Dec-2018.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ 𝐵 ≀ 𝑅𝐴)) | ||
Theorem | brcoels 35682* | 𝐵 and 𝐶 are coelements : a binary relation. (Contributed by Peter Mazsa, 14-Jan-2020.) (Revised by Peter Mazsa, 5-Oct-2021.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∼ 𝐴𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝐵 ∈ 𝑢 ∧ 𝐶 ∈ 𝑢))) | ||
Theorem | cocossss 35683* | Two ways of saying that cosets by cosets by 𝑅 is a subclass. (Contributed by Peter Mazsa, 17-Sep-2021.) |
⊢ ( ≀ ≀ 𝑅 ⊆ 𝑆 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧)) | ||
Theorem | cnvcosseq 35684 | The converse of cosets by 𝑅 are cosets by 𝑅. (Contributed by Peter Mazsa, 3-May-2019.) |
⊢ ◡ ≀ 𝑅 = ≀ 𝑅 | ||
Theorem | br2coss 35685 | Cosets by ≀ 𝑅 binary relation. (Contributed by Peter Mazsa, 25-Aug-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅)) | ||
Theorem | br1cossres 35686* | 𝐵 and 𝐶 are cosets by a restriction: a binary relation. (Contributed by Peter Mazsa, 30-Dec-2018.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ↾ 𝐴)𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑅𝐶))) | ||
Theorem | br1cossres2 35687* | 𝐵 and 𝐶 are cosets by a restriction: a binary relation. (Contributed by Peter Mazsa, 3-Jan-2018.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ↾ 𝐴)𝐶 ↔ ∃𝑥 ∈ 𝐴 (𝐵 ∈ [𝑥]𝑅 ∧ 𝐶 ∈ [𝑥]𝑅))) | ||
Theorem | relbrcoss 35688* | 𝐴 and 𝐵 are cosets by relation 𝑅: a binary relation. (Contributed by Peter Mazsa, 22-Apr-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Rel 𝑅 → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅 ∧ 𝐵 ∈ [𝑥]𝑅)))) | ||
Theorem | br1cossinres 35689* | 𝐵 and 𝐶 are cosets by an intersection with a restriction: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (𝑆 ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶)))) | ||
Theorem | br1cossxrnres 35690* | 〈𝐵, 𝐶〉 and 〈𝐷, 𝐸〉 are cosets by an intersection with a restriction: a binary relation. (Contributed by Peter Mazsa, 8-Jun-2021.) |
⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷)))) | ||
Theorem | br1cossinidres 35691* | 𝐵 and 𝐶 are cosets by an intersection with the restricted identity class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐶)))) | ||
Theorem | br1cossincnvepres 35692* | 𝐵 and 𝐶 are cosets by an intersection with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (◡ E ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶)))) | ||
Theorem | br1cossxrnidres 35693* | 〈𝐵, 𝐶〉 and 〈𝐷, 𝐸〉 are cosets by a range Cartesian product with the restricted identity class: a binary relation. (Contributed by Peter Mazsa, 8-Jun-2021.) |
⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ ( I ↾ 𝐴))〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐸 ∧ 𝑢𝑅𝐷)))) | ||
Theorem | br1cossxrncnvepres 35694* | 〈𝐵, 𝐶〉 and 〈𝐷, 𝐸〉 are cosets by a range Cartesian product with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 12-May-2021.) |
⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ∈ 𝑢 ∧ 𝑢𝑅𝐷)))) | ||
Theorem | dmcoss3 35695 | The domain of cosets is the domain of converse. (Contributed by Peter Mazsa, 4-Jan-2019.) |
⊢ dom ≀ 𝑅 = dom ◡𝑅 | ||
Theorem | dmcoss2 35696 | The domain of cosets is the range. (Contributed by Peter Mazsa, 27-Dec-2018.) |
⊢ dom ≀ 𝑅 = ran 𝑅 | ||
Theorem | rncossdmcoss 35697 | The range of cosets is the domain of them (this should be rncoss 5845 but there exists a theorem with this name already). (Contributed by Peter Mazsa, 12-Dec-2019.) |
⊢ ran ≀ 𝑅 = dom ≀ 𝑅 | ||
Theorem | dm1cosscnvepres 35698 | The domain of cosets of the restricted converse epsilon relation is the union of the restriction. (Contributed by Peter Mazsa, 18-May-2019.) (Revised by Peter Mazsa, 26-Sep-2021.) |
⊢ dom ≀ (◡ E ↾ 𝐴) = ∪ 𝐴 | ||
Theorem | dmcoels 35699 | The domain of coelements in 𝐴 is the union of 𝐴. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Peter Mazsa, 5-Apr-2018.) (Revised by Peter Mazsa, 26-Sep-2021.) |
⊢ dom ∼ 𝐴 = ∪ 𝐴 | ||
Theorem | eldmcoss 35700* | Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |