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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elecres 37601 | Elementhood in the restricted coset of 𝐵. (Contributed by Peter Mazsa, 21-Sep-2018.) |
⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ [𝐵](𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶))) | ||
Theorem | ecres 37602* | Restricted coset of 𝐵. (Contributed by Peter Mazsa, 9-Dec-2018.) |
⊢ [𝐵](𝑅 ↾ 𝐴) = {𝑥 ∣ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝑥)} | ||
Theorem | ecres2 37603 | The restricted coset of 𝐵 when 𝐵 is an element of the restriction. (Contributed by Peter Mazsa, 16-Oct-2018.) |
⊢ (𝐵 ∈ 𝐴 → [𝐵](𝑅 ↾ 𝐴) = [𝐵]𝑅) | ||
Theorem | eccnvepres 37604* | Restricted converse epsilon coset of 𝐵. (Contributed by Peter Mazsa, 11-Feb-2018.) (Revised by Peter Mazsa, 21-Oct-2021.) |
⊢ (𝐵 ∈ 𝑉 → [𝐵](◡ E ↾ 𝐴) = {𝑥 ∈ 𝐵 ∣ 𝐵 ∈ 𝐴}) | ||
Theorem | eleccnvep 37605 | Elementhood in the converse epsilon coset of 𝐴 is elementhood in 𝐴. (Contributed by Peter Mazsa, 27-Jan-2019.) |
⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ [𝐴]◡ E ↔ 𝐵 ∈ 𝐴)) | ||
Theorem | eccnvep 37606 | The converse epsilon coset of a set is the set. (Contributed by Peter Mazsa, 27-Jan-2019.) |
⊢ (𝐴 ∈ 𝑉 → [𝐴]◡ E = 𝐴) | ||
Theorem | extep 37607 | Property of epsilon relation, see also extid 37635, extssr 37835 and the comment of df-ssr 37824. (Contributed by Peter Mazsa, 10-Jul-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ E = [𝐵]◡ E ↔ 𝐴 = 𝐵)) | ||
Theorem | disjeccnvep 37608 | Property of the epsilon relation. (Contributed by Peter Mazsa, 27-Apr-2020.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]◡ E ∩ [𝐵]◡ E ) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅)) | ||
Theorem | eccnvepres2 37609 | The restricted converse epsilon coset of an element of the restriction is the element itself. (Contributed by Peter Mazsa, 16-Jul-2019.) |
⊢ (𝐵 ∈ 𝐴 → [𝐵](◡ E ↾ 𝐴) = 𝐵) | ||
Theorem | eccnvepres3 37610 | Condition for a restricted converse epsilon coset of a set to be the set itself. (Contributed by Peter Mazsa, 11-May-2021.) |
⊢ (𝐵 ∈ dom (◡ E ↾ 𝐴) → [𝐵](◡ E ↾ 𝐴) = 𝐵) | ||
Theorem | eldmqsres 37611* | Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 21-Aug-2020.) |
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 (∃𝑥 𝑥 ∈ [𝑢]𝑅 ∧ 𝐵 = [𝑢]𝑅))) | ||
Theorem | eldmqsres2 37612* | Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 22-Aug-2020.) |
⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (dom (𝑅 ↾ 𝐴) / (𝑅 ↾ 𝐴)) ↔ ∃𝑢 ∈ 𝐴 ∃𝑥 ∈ [ 𝑢]𝑅𝐵 = [𝑢]𝑅)) | ||
Theorem | qsss1 37613 | Subclass theorem for quotient sets. (Contributed by Peter Mazsa, 12-Sep-2020.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐴 / 𝐶) ⊆ (𝐵 / 𝐶)) | ||
Theorem | qseq1i 37614 | Equality theorem for quotient set, inference form. (Contributed by Peter Mazsa, 3-Jun-2021.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 / 𝐶) = (𝐵 / 𝐶) | ||
Theorem | qseq1d 37615 | Equality theorem for quotient set, deduction form. (Contributed by Peter Mazsa, 27-May-2021.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐶)) | ||
Theorem | brinxprnres 37616 | Binary relation on a restriction. (Contributed by Peter Mazsa, 2-Jan-2019.) |
⊢ (𝐶 ∈ 𝑉 → (𝐵(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵𝑅𝐶))) | ||
Theorem | inxprnres 37617* | Restriction of a class as a class of ordered pairs. (Contributed by Peter Mazsa, 2-Jan-2019.) |
⊢ (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦)} | ||
Theorem | dfres4 37618 | Alternate definition of the restriction of a class. (Contributed by Peter Mazsa, 2-Jan-2019.) |
⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴))) | ||
Theorem | exan3 37619* | Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑢(𝐴 ∈ [𝑢]𝑅 ∧ 𝐵 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵))) | ||
Theorem | exanres 37620* | Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 2-May-2021.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢(𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))) | ||
Theorem | exanres3 37621* | Equivalent expressions with restricted existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢 ∈ 𝐴 (𝐵 ∈ [𝑢]𝑅 ∧ 𝐶 ∈ [𝑢]𝑆) ↔ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))) | ||
Theorem | exanres2 37622* | Equivalent expressions with existential quantification. (Contributed by Peter Mazsa, 10-Sep-2021.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (∃𝑢(𝑢(𝑅 ↾ 𝐴)𝐵 ∧ 𝑢(𝑆 ↾ 𝐴)𝐶) ↔ ∃𝑢 ∈ 𝐴 (𝐵 ∈ [𝑢]𝑅 ∧ 𝐶 ∈ [𝑢]𝑆))) | ||
Theorem | cnvepres 37623* | Restricted converse epsilon relation as a class of ordered pairs. (Contributed by Peter Mazsa, 10-Feb-2018.) |
⊢ (◡ E ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)} | ||
Theorem | eqrel2 37624* | Equality of relations. (Contributed by Peter Mazsa, 8-Mar-2019.) |
⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))) | ||
Theorem | rncnv 37625 | Range of converse is the domain. (Contributed by Peter Mazsa, 12-Feb-2018.) |
⊢ ran ◡𝐴 = dom 𝐴 | ||
Theorem | dfdm6 37626* | Alternate definition of domain. (Contributed by Peter Mazsa, 2-Mar-2018.) |
⊢ dom 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅} | ||
Theorem | dfrn6 37627* | Alternate definition of range. (Contributed by Peter Mazsa, 1-Aug-2018.) |
⊢ ran 𝑅 = {𝑥 ∣ [𝑥]◡𝑅 ≠ ∅} | ||
Theorem | rncnvepres 37628 | The range of the restricted converse epsilon is the union of the restriction. (Contributed by Peter Mazsa, 11-Feb-2018.) (Revised by Peter Mazsa, 26-Sep-2021.) |
⊢ ran (◡ E ↾ 𝐴) = ∪ 𝐴 | ||
Theorem | dmecd 37629 | Equality of the coset of 𝐵 and the coset of 𝐶 implies equivalence of domain elementhood (equivalence is not necessary as opposed to ereldm 8746). (Contributed by Peter Mazsa, 9-Oct-2018.) |
⊢ (𝜑 → dom 𝑅 = 𝐴) & ⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅) ⇒ ⊢ (𝜑 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | ||
Theorem | dmec2d 37630 | Equality of the coset of 𝐵 and the coset of 𝐶 implies equivalence of domain elementhood (equivalence is not necessary as opposed to ereldm 8746). (Contributed by Peter Mazsa, 12-Oct-2018.) |
⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅) ⇒ ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐶 ∈ dom 𝑅)) | ||
Theorem | brid 37631 | Property of the identity binary relation. (Contributed by Peter Mazsa, 18-Dec-2021.) |
⊢ (𝐴 I 𝐵 ↔ 𝐵 I 𝐴) | ||
Theorem | ideq2 37632 | For sets, the identity binary relation is the same as equality. (Contributed by Peter Mazsa, 24-Jun-2020.) (Revised by Peter Mazsa, 18-Dec-2021.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | idresssidinxp 37633 | Condition for the identity restriction to be a subclass of identity intersection with a Cartesian product. (Contributed by Peter Mazsa, 19-Jul-2018.) |
⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵))) | ||
Theorem | idreseqidinxp 37634 | Condition for the identity restriction to be equal to the identity intersection with a Cartesian product. (Contributed by Peter Mazsa, 19-Jul-2018.) |
⊢ (𝐴 ⊆ 𝐵 → ( I ∩ (𝐴 × 𝐵)) = ( I ↾ 𝐴)) | ||
Theorem | extid 37635 | Property of identity relation, see also extep 37607, extssr 37835 and the comment of df-ssr 37824. (Contributed by Peter Mazsa, 5-Jul-2019.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵)) | ||
Theorem | inxpss 37636* | Two ways to say that an intersection with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 16-Jul-2019.) |
⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ 𝑆 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥𝑆𝑦)) | ||
Theorem | idinxpss 37637* | Two ways to say that an intersection of the identity relation with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 16-Jul-2019.) |
⊢ (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 → 𝑥𝑅𝑦)) | ||
Theorem | ref5 37638* | Two ways to say that an intersection of the identity relation with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 12-Dec-2023.) |
⊢ (( I ∩ (𝐴 × 𝐵)) ⊆ 𝑅 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐵)𝑥𝑅𝑥) | ||
Theorem | inxpss3 37639* | Two ways to say that an intersection with a Cartesian product is a subclass (see also inxpss 37636). (Contributed by Peter Mazsa, 8-Mar-2019.) |
⊢ (∀𝑥∀𝑦(𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 → 𝑥(𝑆 ∩ (𝐴 × 𝐵))𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥𝑆𝑦)) | ||
Theorem | inxpss2 37640* | Two ways to say that intersections with Cartesian products are in a subclass relation. (Contributed by Peter Mazsa, 8-Mar-2019.) |
⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑆 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥𝑆𝑦)) | ||
Theorem | inxpssidinxp 37641* | Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 37640. (Contributed by Peter Mazsa, 4-Jul-2019.) |
⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⊆ ( I ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 → 𝑥 = 𝑦)) | ||
Theorem | idinxpssinxp 37642* | Two ways to say that intersections with Cartesian products are in a subclass relation, special case of inxpss2 37640. (Contributed by Peter Mazsa, 6-Mar-2019.) |
⊢ (( I ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 → 𝑥𝑅𝑦)) | ||
Theorem | idinxpssinxp2 37643* | Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product. (Contributed by Peter Mazsa, 4-Mar-2019.) (Proof modification is discouraged.) |
⊢ (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) | ||
Theorem | idinxpssinxp3 37644 | Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product. (Contributed by Peter Mazsa, 16-Mar-2019.) (Proof modification is discouraged.) |
⊢ (( I ∩ (𝐴 × 𝐴)) ⊆ (𝑅 ∩ (𝐴 × 𝐴)) ↔ ( I ↾ 𝐴) ⊆ 𝑅) | ||
Theorem | idinxpssinxp4 37645* | Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product (see also idinxpssinxp2 37643). (Contributed by Peter Mazsa, 8-Mar-2019.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥) | ||
Theorem | relcnveq3 37646* | Two ways of saying a relation is symmetric. (Contributed by FL, 31-Aug-2009.) |
⊢ (Rel 𝑅 → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) | ||
Theorem | relcnveq 37647 | Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 23-Aug-2018.) |
⊢ (Rel 𝑅 → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅)) | ||
Theorem | relcnveq2 37648* | 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 37649* | 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 37650 | Simplification of a special quotient set. (Contributed by Peter Mazsa, 2-Sep-2020.) |
⊢ (𝐴 / (𝑅 ↾ 𝐴)) = (𝐴 / 𝑅) | ||
Theorem | n0elqs 37651 | 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 37652 | 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 37653 | 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 37654 | The union of a quotient set, like uniqs 8766 but with a weaker antecedent: only the restricion of 𝑅 by 𝐴 needs to be a set, not 𝑅 itself, see e.g. cnvepima 37662. (Contributed by Peter Mazsa, 20-Jun-2019.) |
⊢ ((𝑅 ↾ 𝐴) ∈ 𝑉 → ∪ (𝐴 / 𝑅) = (𝑅 “ 𝐴)) | ||
Theorem | imaexALTV 37655 | Existence of an image of a class. Theorem 3.17 of [Monk1] p. 39. (cf. imaexg 7899) with weakened antecedent: only the restricion of 𝐴 by a set needs to be a set, not 𝐴 itself, see e.g. cnvepimaex 37661. (Contributed by Peter Mazsa, 22-Feb-2023.) (Proof modification is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∨ ((𝐴 ↾ 𝐵) ∈ 𝑊 ∧ 𝐵 ∈ 𝑋)) → (𝐴 “ 𝐵) ∈ V) | ||
Theorem | ecexALTV 37656 | Existence of a coset, like ecexg 8702 but with a weaker antecedent: only the restricion of 𝑅 by the singleton of 𝐴 needs to be a set, not 𝑅 itself, see e.g. eccnvepex 37660. (Contributed by Peter Mazsa, 22-Feb-2023.) |
⊢ ((𝑅 ↾ {𝐴}) ∈ 𝑉 → [𝐴]𝑅 ∈ V) | ||
Theorem | rnresequniqs 37657 | 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 37658 | 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 37659 | Sethood condition for the restricted converse epsilon relation. (Contributed by Peter Mazsa, 24-Sep-2018.) |
⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V) | ||
Theorem | eccnvepex 37660 | The converse epsilon coset exists. (Contributed by Peter Mazsa, 22-Mar-2023.) |
⊢ [𝐴]◡ E ∈ V | ||
Theorem | cnvepimaex 37661 | The image of converse epsilon exists, proof via imaexALTV 37655 (see also cnvepima 37662 and uniexg 7723 for alternate way). (Contributed by Peter Mazsa, 22-Mar-2023.) |
⊢ (𝐴 ∈ 𝑉 → (◡ E “ 𝐴) ∈ V) | ||
Theorem | cnvepima 37662 | The image of converse epsilon. (Contributed by Peter Mazsa, 22-Mar-2023.) |
⊢ (𝐴 ∈ 𝑉 → (◡ E “ 𝐴) = ∪ 𝐴) | ||
Theorem | inex3 37663 | 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 37664 | 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 37665 | Converting a class constant definition by restriction (like df-ers 37989 or df-parts 38091) into a binary relation. (Contributed by Peter Mazsa, 1-Oct-2018.) |
⊢ 𝑅 = (𝑆 ↾ 𝐶) ⇒ ⊢ (𝐵 ∈ 𝑉 → (𝐴𝑅𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐴𝑆𝐵))) | ||
Theorem | brrabga 37666* | 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 37667* | 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 37668 | 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 37669 | 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 37670* | Elementhood in a domain of a converse. (Contributed by Peter Mazsa, 25-May-2018.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) | ||
Theorem | dfrel5 37671 | Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 6-Nov-2018.) |
⊢ (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅) | ||
Theorem | dfrel6 37672 | Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 14-Mar-2019.) |
⊢ (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅) | ||
Theorem | cnvresrn 37673 | Converse restricted to range is converse. (Contributed by Peter Mazsa, 3-Sep-2021.) |
⊢ (◡𝑅 ↾ ran 𝑅) = ◡𝑅 | ||
Theorem | relssinxpdmrn 37674 | Subset of restriction, special case. (Contributed by Peter Mazsa, 10-Apr-2023.) |
⊢ (Rel 𝑅 → (𝑅 ⊆ (𝑆 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ 𝑆)) | ||
Theorem | cnvref4 37675 | 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 37676* | 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 37677* | 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 37678* | 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 37679* | 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 37680* | 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 37681* | 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 37682* | 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 37683 | 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 37684* | 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 37685* | 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 37686* | 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 37687 | 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 37688 | "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 2624) 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 37689 | 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 37690* | The law of concretion for a binary relation. Special case of brabga 5524. Version of brabidga 37691 with a disjoint variable condition, which does not require ax-13 2363. (Contributed by Peter Mazsa, 24-Nov-2018.) (Revised by Gino Giotto, 2-Apr-2024.) |
⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} ⇒ ⊢ (𝑥𝑅𝑦 ↔ 𝜑) | ||
Theorem | brabidga 37691 | The law of concretion for a binary relation. Special case of brabga 5524. Usage of this theorem is discouraged because it depends on ax-13 2363, see brabidgaw 37690 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 37692* | Intersection with a Cartesian product. (Contributed by Peter Mazsa, 18-Jul-2019.) |
⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)} | ||
Theorem | opabf 37693 | 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 37694 | The empty-coset of a class is the empty set. (Contributed by Peter Mazsa, 19-May-2019.) |
⊢ [𝐴]∅ = ∅ | ||
Theorem | 0qs 37695 | Quotient set with the empty set. (Contributed by Peter Mazsa, 14-Sep-2019.) |
⊢ (∅ / 𝑅) = ∅ | ||
Theorem | brcnvin 37696 | 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 37697 | Define the range Cartesian product of two classes. Definition from [Holmes] p. 40. Membership in this class is characterized by xrnss3v 37698 and brxrn 37700. This is Scott Fenton's df-txp 35287 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 35287. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ (𝐴 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) | ||
Theorem | xrnss3v 37698 | A range Cartesian product is a subset of the class of ordered triples. This is Scott Fenton's txpss3v 35311 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 35311. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V)) | ||
Theorem | xrnrel 37699 | A range Cartesian product is a relation. This is Scott Fenton's txprel 35312 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 35312. (Contributed by Scott Fenton, 31-Mar-2012.) |
⊢ Rel (𝐴 ⋉ 𝐵) | ||
Theorem | brxrn 37700 | Characterize a ternary relation over a range Cartesian product. Together with xrnss3v 37698, 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))) |
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