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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | idinxpssinxp4 38301* | Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product (see also idinxpssinxp2 38299). (Contributed by Peter Mazsa, 8-Mar-2019.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) | ||
Theorem | relcnveq3 38302* | Two ways of saying a relation is symmetric. (Contributed by FL, 31-Aug-2009.) |
⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) | ||
Theorem | relcnveq 38303 | Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 23-Aug-2018.) |
⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅)) | ||
Theorem | relcnveq2 38304* | Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.) |
⊢ (Rel 𝑅 → (◡𝑅 = 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥))) | ||
Theorem | relcnveq4 38305* | Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 28-Apr-2019.) |
⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥))) | ||
Theorem | qsresid 38306 | Simplification of a special quotient set. (Contributed by Peter Mazsa, 2-Sep-2020.) |
⊢ (𝐴 / (𝑅 ↾ 𝐴)) = (𝐴 / 𝑅) | ||
Theorem | n0elqs 38307 | Two ways of expressing that the empty set is not an element of a quotient set. (Contributed by Peter Mazsa, 5-Dec-2019.) |
⊢ (¬ ∅ ∈ (𝐴 / 𝑅) ↔ 𝐴 ⊆ dom 𝑅) | ||
Theorem | n0elqs2 38308 | Two ways of expressing that the empty set is not an element of a quotient set. (Contributed by Peter Mazsa, 25-Jul-2021.) |
⊢ (¬ ∅ ∈ (𝐴 / 𝑅) ↔ dom (𝑅 ↾ 𝐴) = 𝐴) | ||
Theorem | ecex2 38309 | Condition for a coset to be a set. (Contributed by Peter Mazsa, 4-May-2019.) |
⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → (𝐵 ∈ 𝐴 → [𝐵]𝑅 ∈ V)) | ||
Theorem | uniqsALTV 38310 | The union of a quotient set, like uniqs 8815 but with a weaker antecedent: only the restriction of 𝑅 by 𝐴 needs to be a set, not 𝑅 itself, see e.g. cnvepima 38318. (Contributed by Peter Mazsa, 20-Jun-2019.) |
⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴)) | ||
Theorem | imaexALTV 38311 | Existence of an image of a class. Theorem 3.17 of [Monk1] p. 39. (cf. imaexg 7935) with weakened antecedent: only the restriction of 𝐴 by a set needs to be a set, not 𝐴 itself, see e.g. cnvepimaex 38317. (Contributed by Peter Mazsa, 22-Feb-2023.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∨ ((𝐴 ↾ 𝐵) ∈ 𝑊 ∧ 𝐵 ∈ 𝑋)) → (𝐴 “ 𝐵) ∈ V) | ||
Theorem | ecexALTV 38312 | Existence of a coset, like ecexg 8747 but with a weaker antecedent: only the restriction of 𝑅 by the singleton of 𝐴 needs to be a set, not 𝑅 itself, see e.g. eccnvepex 38316. (Contributed by Peter Mazsa, 22-Feb-2023.) |
⊢ ((𝑅 ↾ {𝐴}) ∈ 𝑉 → [𝐴]𝑅 ∈ V) | ||
Theorem | rnresequniqs 38313 | The range of a restriction is equal to the union of the quotient set. (Contributed by Peter Mazsa, 19-May-2018.) |
⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ran (𝑅 ↾ 𝐴) = ∪ (𝐴 / 𝑅)) | ||
Theorem | n0el2 38314 | Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 31-Jan-2018.) |
⊢ (¬ ∅ ∈ 𝐴 ↔ dom (◡ E ↾ 𝐴) = 𝐴) | ||
Theorem | cnvepresex 38315 | Sethood condition for the restricted converse epsilon relation. (Contributed by Peter Mazsa, 24-Sep-2018.) |
⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V) | ||
Theorem | eccnvepex 38316 | The converse epsilon coset exists. (Contributed by Peter Mazsa, 22-Mar-2023.) |
⊢ [𝐴]◡ E ∈ V | ||
Theorem | cnvepimaex 38317 | The image of converse epsilon exists, proof via imaexALTV 38311 (see also cnvepima 38318 and uniexg 7758 for alternate way). (Contributed by Peter Mazsa, 22-Mar-2023.) |
⊢ (𝐴 ∈ 𝑉 → (◡ E “ 𝐴) ∈ V) | ||
Theorem | cnvepima 38318 | The image of converse epsilon. (Contributed by Peter Mazsa, 22-Mar-2023.) |
⊢ (𝐴 ∈ 𝑉 → (◡ E “ 𝐴) = ∪ 𝐴) | ||
Theorem | inex3 38319 | Sufficient condition for the intersection relation to be a set. (Contributed by Peter Mazsa, 24-Nov-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∩ 𝐵) ∈ V) | ||
Theorem | inxpex 38320 | Sufficient condition for an intersection with a Cartesian product to be a set. (Contributed by Peter Mazsa, 10-May-2019.) |
⊢ ((𝑅 ∈ 𝑊 ∨ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V) | ||
Theorem | eqres 38321 | Converting a class constant definition by restriction (like df-ers 38644 or df-parts 38746) into a binary relation. (Contributed by Peter Mazsa, 1-Oct-2018.) |
⊢ 𝑅 = (𝑆 ↾ 𝐶) ⇒ ⊢ (𝐵 ∈ 𝑉 → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵))) | ||
Theorem | brrabga 38322* | The law of concretion for operation class abstraction. (Contributed by Peter Mazsa, 24-Oct-2022.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) & ⊢ 𝑅 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈𝐴, 𝐵〉𝑅𝐶 ↔ 𝜓)) | ||
Theorem | brcnvrabga 38323* | 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 38324 | 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 38325 | 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 38326* | Elementhood in a domain of a converse. (Contributed by Peter Mazsa, 25-May-2018.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) | ||
Theorem | dfrel5 38327 | Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 6-Nov-2018.) |
⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅) | ||
Theorem | dfrel6 38328 | Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 14-Mar-2019.) |
⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅) | ||
Theorem | cnvresrn 38329 | Converse restricted to range is converse. (Contributed by Peter Mazsa, 3-Sep-2021.) |
⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅 | ||
Theorem | relssinxpdmrn 38330 | Subset of restriction, special case. (Contributed by Peter Mazsa, 10-Apr-2023.) |
⊢ (Rel 𝑅 → (𝑅 ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ 𝑆)) | ||
Theorem | cnvref4 38331 | Two ways to say that a relation is a subclass. (Contributed by Peter Mazsa, 11-Apr-2023.) |
⊢ (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ 𝑆)) | ||
Theorem | cnvref5 38332* | Two ways to say that a relation is a subclass of the identity relation. (Contributed by Peter Mazsa, 26-Jun-2019.) |
⊢ (Rel 𝑅 → (𝑅 ⊆ I ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑥 = 𝑦))) | ||
Theorem | ecin0 38333* | 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 38334* | 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 38335* | 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 38336* | 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 38337* | 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 38338* | 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 38339 | 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 38340* | 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 38341* | 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 38342* | 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 | moeu2 38343 | Uniqueness is equivalent to non-existence or unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by Peter Mazsa, 19-Nov-2024.) |
⊢ (∃*𝑥𝜑 ↔ (¬ ∃𝑥𝜑 ∨ ∃!𝑥𝜑)) | ||
Theorem | mopickr 38344 | "At most one" picks a variable value, eliminating an existential quantifier. The proof begins with references *2.21 (pm2.21 123) and *14.26 (eupickbi 2633) from [WhiteheadRussell] p. 104 and p. 183. (Contributed by Peter Mazsa, 18-Nov-2024.) (Proof modification is discouraged.) |
⊢ ((∃*𝑥𝜓 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜓 → 𝜑)) | ||
Theorem | moantr 38345 | 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 38346* | The law of concretion for a binary relation. Special case of brabga 5543. Version of brabidga 38347 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by Peter Mazsa, 24-Nov-2018.) (Revised by GG, 2-Apr-2024.) |
⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ (𝑥𝑅𝑦 ↔ 𝜑) | ||
Theorem | brabidga 38347 | The law of concretion for a binary relation. Special case of brabga 5543. Usage of this theorem is discouraged because it depends on ax-13 2374, see brabidgaw 38346 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 38348* | Intersection with a Cartesian product. (Contributed by Peter Mazsa, 18-Jul-2019.) |
⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)} | ||
Theorem | opabf 38349 | 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 38350 | The empty-coset of a class is the empty set. (Contributed by Peter Mazsa, 19-May-2019.) |
⊢ [𝐴]∅ = ∅ | ||
Theorem | brcnvin 38351 | Intersection with a converse, binary relation. (Contributed by Peter Mazsa, 24-Mar-2024.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ ◡𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑆𝐴))) | ||
Definition | df-xrn 38352 | Define the range Cartesian product of two classes. Definition from [Holmes] p. 40. Membership in this class is characterized by xrnss3v 38353 and brxrn 38355. This is Scott Fenton's df-txp 35835 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 35835. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ (𝐴 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) | ||
Theorem | xrnss3v 38353 | A range Cartesian product is a subset of the class of ordered triples. This is Scott Fenton's txpss3v 35859 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 35859. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V)) | ||
Theorem | xrnrel 38354 | A range Cartesian product is a relation. This is Scott Fenton's txprel 35860 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 35860. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ Rel (𝐴 ⋉ 𝐵) | ||
Theorem | brxrn 38355 | Characterize a ternary relation over a range Cartesian product. Together with xrnss3v 38353, 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 38356* | 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 38357* | Alternate definition of the range Cartesian product. (Contributed by Peter Mazsa, 20-Feb-2022.) |
⊢ (𝑅 ⋉ 𝑆) = ◡{〈〈𝑥, 𝑦〉, 𝑢〉 ∣ (𝑢𝑅𝑥 ∧ 𝑢𝑆𝑦)} | ||
Theorem | xrneq1 38358 | Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.) |
⊢ (𝐴 = 𝐵 → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶)) | ||
Theorem | xrneq1i 38359 | Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐶) | ||
Theorem | xrneq1d 38360 | 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 38361 | Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.) |
⊢ (𝐴 = 𝐵 → (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵)) | ||
Theorem | xrneq2i 38362 | Equality theorem for the range Cartesian product, inference form. (Contributed by Peter Mazsa, 16-Dec-2020.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ⋉ 𝐴) = (𝐶 ⋉ 𝐵) | ||
Theorem | xrneq2d 38363 | 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 38364 | Equality theorem for the range Cartesian product. (Contributed by Peter Mazsa, 16-Dec-2020.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⋉ 𝐶) = (𝐵 ⋉ 𝐷)) | ||
Theorem | xrneq12i 38365 | 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 38366 | 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 38367* | 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 38368* | 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 | disjressuc2 38369* | Double restricted quantification over the union of a set and its singleton. (Contributed by Peter Mazsa, 22-Aug-2023.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢 ∈ 𝐴 ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))) | ||
Theorem | disjecxrn 38370 | Two ways of saying that (𝑅 ⋉ 𝑆)-cosets are disjoint. (Contributed by Peter Mazsa, 19-Jun-2020.) (Revised by Peter Mazsa, 21-Aug-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = ∅ ↔ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴]𝑆 ∩ [𝐵]𝑆) = ∅))) | ||
Theorem | disjecxrncnvep 38371 | Two ways of saying that cosets are disjoint, special case of disjecxrn 38370. (Contributed by Peter Mazsa, 12-Jul-2020.) (Revised by Peter Mazsa, 25-Aug-2023.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ ◡ E ) ∩ [𝐵](𝑅 ⋉ ◡ E )) = ∅ ↔ ((𝐴 ∩ 𝐵) = ∅ ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))) | ||
Theorem | disjsuc2 38372* | Double restricted quantification over the union of a set and its singleton. (Contributed by Peter Mazsa, 22-Aug-2023.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅) ↔ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅) ∧ ∀𝑢 ∈ 𝐴 ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))) | ||
Theorem | xrninxp 38373* | 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 38374* | 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 38375 | 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 38376 | 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 38377* | 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 38378* | 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 38379* | 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 38380* | 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 38381* | 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 38382* | 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 38383 | Two ways to express restriction of range Cartesian product, see also xrnres2 38384, xrnres3 38385. (Contributed by Peter Mazsa, 5-Jun-2021.) |
⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ 𝑆) | ||
Theorem | xrnres2 38384 | Two ways to express restriction of range Cartesian product, see also xrnres 38383, xrnres3 38385. (Contributed by Peter Mazsa, 6-Sep-2021.) |
⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴)) | ||
Theorem | xrnres3 38385 | Two ways to express restriction of range Cartesian product, see also xrnres 38383, xrnres2 38384. (Contributed by Peter Mazsa, 28-Mar-2020.) |
⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ((𝑅 ↾ 𝐴) ⋉ (𝑆 ↾ 𝐴)) | ||
Theorem | xrnres4 38386 | 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 38387 | 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 38388 | 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 38389 | 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 38390 | 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 38391 | 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 38392* |
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 38394 and the comment of dfec2 8746). 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 38832). Without the definition of ≀ 𝑅 we should have to relate the right side of these theorems to a composition of a converse (cf. dfcoss3 38395) or to the range of a range Cartesian product of classes (cf. dfcoss4 38396), which would make the theorems complicated and confusing. Alternate definition is dfcoss2 38394. Technically, we can define it via composition (dfcoss3 38395) or as the range of a range Cartesian product (dfcoss4 38396), but neither of these definitions reveal directly how the cosets by 𝑅 relate to each other. We define functions (df-funsALTV 38662, df-funALTV 38663) and disjoints (dfdisjs 38689, dfdisjs2 38690, df-disjALTV 38686, dfdisjALTV2 38695) with the help of it as well. (Contributed by Peter Mazsa, 9-Jan-2018.) |
⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | ||
Definition | df-coels 38393 | Define the class of coelements on the class 𝐴, see also the alternate definition dfcoels 38411. Possible definitions are the special cases of dfcoss3 38395 and dfcoss4 38396. (Contributed by Peter Mazsa, 20-Nov-2019.) |
⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | ||
Theorem | dfcoss2 38394* | 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 8746). 𝑅 is usually a relation. (Contributed by Peter Mazsa, 16-Jan-2018.) |
⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)} | ||
Theorem | dfcoss3 38395 | Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 38392). (Contributed by Peter Mazsa, 27-Dec-2018.) |
⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅) | ||
Theorem | dfcoss4 38396 | Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 38392). (Contributed by Peter Mazsa, 12-Jul-2021.) |
⊢ ≀ 𝑅 = ran (𝑅 ⋉ 𝑅) | ||
Theorem | cosscnv 38397* | Class of cosets by the converse of 𝑅 (Contributed by Peter Mazsa, 17-Jun-2020.) |
⊢ ≀ ◡𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)} | ||
Theorem | coss1cnvres 38398* | Class of cosets by the converse of a restriction. (Contributed by Peter Mazsa, 8-Jun-2020.) |
⊢ ≀ ◡(𝑅 ↾ 𝐴) = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))} | ||
Theorem | coss2cnvepres 38399* | Special case of coss1cnvres 38398. (Contributed by Peter Mazsa, 8-Jun-2020.) |
⊢ ≀ ◡(◡ E ↾ 𝐴) = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣))} | ||
Theorem | cossex 38400 | 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) |
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