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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cosscnvssid3 36601* | Equivalent expressions for the class of cosets by the converse of 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 28-Jul-2021.) |
⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑢∀𝑣∀𝑥((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → 𝑢 = 𝑣)) | ||
Theorem | cosscnvssid4 36602* | Equivalent expressions for the class of cosets by the converse of 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 31-Aug-2021.) |
⊢ ( ≀ ◡𝑅 ⊆ I ↔ ∀𝑥∃*𝑢 𝑢𝑅𝑥) | ||
Theorem | cosscnvssid5 36603* | Equivalent expressions for the class of cosets by the converse of the relation 𝑅 to be a subset of the identity class. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ (( ≀ ◡𝑅 ⊆ I ∧ Rel 𝑅) ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅)) | ||
Theorem | coss0 36604 | Cosets by the empty set are the empty set. (Contributed by Peter Mazsa, 22-Oct-2019.) |
⊢ ≀ ∅ = ∅ | ||
Theorem | cossid 36605 | Cosets by the identity relation are the identity relation. (Contributed by Peter Mazsa, 16-Jan-2019.) |
⊢ ≀ I = I | ||
Theorem | cosscnvid 36606 | Cosets by the converse identity relation are the identity relation. (Contributed by Peter Mazsa, 27-Sep-2021.) |
⊢ ≀ ◡ I = I | ||
Theorem | trcoss 36607* | Sufficient condition for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 26-Dec-2018.) |
⊢ (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧)) | ||
Theorem | eleccossin 36608 | Two ways of saying that the coset of 𝐴 and the coset of 𝐶 have the common element 𝐵. (Contributed by Peter Mazsa, 15-Oct-2021.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ∈ ([𝐴] ≀ 𝑅 ∩ [𝐶] ≀ 𝑅) ↔ (𝐴 ≀ 𝑅𝐵 ∧ 𝐵 ≀ 𝑅𝐶))) | ||
Theorem | trcoss2 36609* | Equivalent expressions for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 4-Jul-2020.) (Revised by Peter Mazsa, 16-Oct-2021.) |
⊢ (∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧) ↔ ∀𝑥∀𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]◡𝑅 ∩ [𝑧]◡𝑅) ≠ ∅)) | ||
Definition | df-rels 36610 |
Define the relations class. Proper class relations (like I, see
reli 5738) are not elements of it. The element of this
class and the
relation predicate are the same when 𝑅 is a set (see elrelsrel 36612).
The class of relations is a great tool we can use when we define classes of different relations as nullary class constants as required by the 2. point in our Guidelines https://us.metamath.org/mpeuni/mathbox.html 36612. When we want to define a specific class of relations as a nullary class constant, the appropriate method is the following: 1. We define the specific nullary class constant for general sets (see e.g. df-refs 36635), then 2. we get the required class of relations by the intersection of the class of general sets above with the class of relations df-rels 36610 (see df-refrels 36636 and the resulting dfrefrels2 36638 and dfrefrels3 36639). 3. Finally, in order to be able to work with proper classes (like iprc 7769) as well, we define the predicate of the relation (see df-refrel 36637) so that it is true for the relevant proper classes (see refrelid 36646), and that the element of the class of the required relations (e.g. elrefrels3 36643) and this predicate are the same in case of sets (see elrefrelsrel 36644). (Contributed by Peter Mazsa, 13-Jun-2018.) |
⊢ Rels = 𝒫 (V × V) | ||
Theorem | elrels2 36611 | The element of the relations class (df-rels 36610) and the relation predicate (df-rel 5597) are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 14-Jun-2018.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V))) | ||
Theorem | elrelsrel 36612 | The element of the relations class (df-rels 36610) and the relation predicate are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 24-Nov-2018.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅)) | ||
Theorem | elrelsrelim 36613 | The element of the relations class is a relation. (Contributed by Peter Mazsa, 20-Jul-2019.) |
⊢ (𝑅 ∈ Rels → Rel 𝑅) | ||
Theorem | elrels5 36614 | Equivalent expressions for an element of the relations class. (Contributed by Peter Mazsa, 21-Jul-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ (𝑅 ↾ dom 𝑅) = 𝑅)) | ||
Theorem | elrels6 36615 | Equivalent expressions for an element of the relations class. (Contributed by Peter Mazsa, 21-Jul-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)) | ||
Theorem | elrelscnveq3 36616* | Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.) |
⊢ (𝑅 ∈ Rels → (𝑅 = ◡𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥))) | ||
Theorem | elrelscnveq 36617 | Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.) |
⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ◡𝑅 = 𝑅)) | ||
Theorem | elrelscnveq2 36618* | Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.) |
⊢ (𝑅 ∈ Rels → (◡𝑅 = 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥))) | ||
Theorem | elrelscnveq4 36619* | Two ways of saying a relation is symmetric. (Contributed by Peter Mazsa, 22-Aug-2021.) |
⊢ (𝑅 ∈ Rels → (◡𝑅 ⊆ 𝑅 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥))) | ||
Theorem | cnvelrels 36620 | The converse of a set is an element of the class of relations. (Contributed by Peter Mazsa, 18-Aug-2019.) |
⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ Rels ) | ||
Theorem | cosselrels 36621 | Cosets of sets are elements of the relations class. Implies ⊢ (𝑅 ∈ Rels → ≀ 𝑅 ∈ Rels ). (Contributed by Peter Mazsa, 25-Aug-2021.) |
⊢ (𝐴 ∈ 𝑉 → ≀ 𝐴 ∈ Rels ) | ||
Theorem | cosscnvelrels 36622 | Cosets of converse sets are elements of the relations class. (Contributed by Peter Mazsa, 31-Aug-2021.) |
⊢ (𝐴 ∈ 𝑉 → ≀ ◡𝐴 ∈ Rels ) | ||
Definition | df-ssr 36623* |
Define the subsets class or the class of subset relations. Similar to
definitions of epsilon relation (df-eprel 5496) and identity relation
(df-id 5490) classes. Subset relation class and Scott
Fenton's subset
class df-sset 34167 are the same: S = SSet (compare dfssr2 36624 with
df-sset 34167), the only reason we do not use dfssr2 36624 as the base
definition of the subsets class is the way we defined the epsilon
relation and the identity relation classes.
The binary relation on the class of subsets and the subclass relationship (df-ss 3905) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set, see brssr 36626. Yet in general we use the subclass relation 𝐴 ⊆ 𝐵 both for classes and for sets, see the comment of df-ss 3905. The only exception (aside from directly investigating the class S e.g. in relssr 36625 or in extssr 36634) is when we have a specific purpose with its usage, like in case of df-refs 36635 versus df-cnvrefs 36648, where we need S to define the class of reflexive sets in order to be able to define the class of converse reflexive sets with the help of the converse of S. The subsets class S has another place in set.mm as well: if we define extensional relation based on the common property in extid 36453, extep 36426 and extssr 36634, then "extrelssr" " |- ExtRel S " is a theorem along with "extrelep" " |- ExtRel E " and "extrelid" " |- ExtRel I " . (Contributed by Peter Mazsa, 25-Jul-2019.) |
⊢ S = {〈𝑥, 𝑦〉 ∣ 𝑥 ⊆ 𝑦} | ||
Theorem | dfssr2 36624 | Alternate definition of the subset relation. (Contributed by Peter Mazsa, 9-Aug-2021.) |
⊢ S = ((V × V) ∖ ran ( E ⋉ (V ∖ E ))) | ||
Theorem | relssr 36625 | The subset relation is a relation. (Contributed by Peter Mazsa, 1-Aug-2019.) |
⊢ Rel S | ||
Theorem | brssr 36626 | The subset relation and subclass relationship (df-ss 3905) are the same, that is, (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵) when 𝐵 is a set. (Contributed by Peter Mazsa, 31-Jul-2019.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 S 𝐵 ↔ 𝐴 ⊆ 𝐵)) | ||
Theorem | brssrid 36627 | Any set is a subset of itself. (Contributed by Peter Mazsa, 1-Aug-2019.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 S 𝐴) | ||
Theorem | issetssr 36628 | Two ways of expressing set existence. (Contributed by Peter Mazsa, 1-Aug-2019.) |
⊢ (𝐴 ∈ V ↔ 𝐴 S 𝐴) | ||
Theorem | brssrres 36629 | Restricted subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.) |
⊢ (𝐶 ∈ 𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶))) | ||
Theorem | br1cnvssrres 36630 | Restricted converse subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.) |
⊢ (𝐵 ∈ 𝑉 → (𝐵◡( S ↾ 𝐴)𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐵))) | ||
Theorem | brcnvssr 36631 | The converse of a subset relation swaps arguments. (Contributed by Peter Mazsa, 1-Aug-2019.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴◡ S 𝐵 ↔ 𝐵 ⊆ 𝐴)) | ||
Theorem | brcnvssrid 36632 | Any set is a converse subset of itself. (Contributed by Peter Mazsa, 9-Jun-2021.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴◡ S 𝐴) | ||
Theorem | br1cossxrncnvssrres 36633* | 〈𝐵, 𝐶〉 and 〈𝐷, 𝐸〉 are cosets by range Cartesian product with restricted converse subsets class: a binary relation. (Contributed by Peter Mazsa, 9-Jun-2021.) |
⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ (◡ S ↾ 𝐴))〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ⊆ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ⊆ 𝑢 ∧ 𝑢𝑅𝐷)))) | ||
Theorem | extssr 36634 | Property of subset relation, see also extid 36453, extep 36426 and the comment of df-ssr 36623. (Contributed by Peter Mazsa, 10-Jul-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ S = [𝐵]◡ S ↔ 𝐴 = 𝐵)) | ||
Definition | df-refs 36635 |
Define the class of all reflexive sets. It is used only by df-refrels 36636.
We use subset relation S (df-ssr 36623) here to be able to define
converse reflexivity (df-cnvrefs 36648), see also the comment of df-ssr 36623.
The elements of this class are not necessarily relations (versus
df-refrels 36636).
Note the similarity of Definitions df-refs 36635, df-syms 36663 and df-trs 36693, cf. comments of dfrefrels2 36638. (Contributed by Peter Mazsa, 19-Jul-2019.) |
⊢ Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))} | ||
Definition | df-refrels 36636 |
Define the class of reflexive relations. This is practically dfrefrels2 36638
(which reveals that RefRels can not include proper
classes like I
as is elements, see comments of dfrefrels2 36638).
Another alternative definition is dfrefrels3 36639. The element of this class and the reflexive relation predicate (df-refrel 36637) are the same, that is, (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝐴 is a set, see elrefrelsrel 36644. This definition is similar to the definitions of the classes of symmetric (df-symrels 36664) and transitive (df-trrels 36694) relations. (Contributed by Peter Mazsa, 7-Jul-2019.) |
⊢ RefRels = ( Refs ∩ Rels ) | ||
Definition | df-refrel 36637 | Define the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.) This is a surprising definition, see the comment of dfrefrel3 36641. Alternate definitions are dfrefrel2 36640 and dfrefrel3 36641. For sets, being an element of the class of reflexive relations (df-refrels 36636) is equivalent to satisfying the reflexive relation predicate, that is (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝑅 is a set, see elrefrelsrel 36644. (Contributed by Peter Mazsa, 16-Jul-2021.) |
⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)) | ||
Theorem | dfrefrels2 36638 |
Alternate definition of the class of reflexive relations. This is a 0-ary
class constant, which is recommended for definitions (see the 1.
Guideline at https://us.metamath.org/ileuni/mathbox.html).
Proper
classes (like I, see iprc 7769)
are not elements of this (or any)
class: if a class is an element of another class, it is not a proper class
but a set, see elex 3451. So if we use 0-ary constant classes as our
main
definitions, they are valid only for sets, not for proper classes. For
proper classes we use predicate-type definitions like df-refrel 36637. See
also the comment of df-rels 36610.
Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 36638, it keeps restriction of I: this is why the very similar definitions df-refs 36635, df-syms 36663 and df-trs 36693 diverge when we switch from (general) sets to relations in dfrefrels2 36638, dfsymrels2 36666 and dftrrels2 36696. (Contributed by Peter Mazsa, 20-Jul-2019.) |
⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} | ||
Theorem | dfrefrels3 36639* | Alternate definition of the class of reflexive relations. (Contributed by Peter Mazsa, 8-Jul-2019.) |
⊢ RefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦)} | ||
Theorem | dfrefrel2 36640 | Alternate definition of the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.) |
⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅)) | ||
Theorem | dfrefrel3 36641* |
Alternate definition of the reflexive relation predicate. A relation is
reflexive iff: for all elements on its domain and range, if an element
of its domain is the same as an element of its range, then there is the
relation between them.
Note that this is definitely not the definition we are accustomed to, like e.g. idref 7027 / idrefALT 6023 or df-reflexive 46481 ⊢ (𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴𝑥𝑅𝑥)). It turns out that the not-surprising definition which contains ∀𝑥 ∈ dom 𝑟𝑥𝑟𝑥 needs symmetry as well, see refsymrels3 36687. Only when this symmetry condition holds, like in case of equivalence relations, see dfeqvrels3 36709, can we write the traditional form ∀𝑥 ∈ dom 𝑟𝑥𝑟𝑥 for reflexive relations. For the special case with square Cartesian product when the two forms are equivalent see idinxpssinxp4 36462 where ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 = 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴𝑥𝑅𝑥). See also similar definition of the converse reflexive relations class dfcnvrefrel3 36654. (Contributed by Peter Mazsa, 8-Jul-2019.) |
⊢ ( RefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) ∧ Rel 𝑅)) | ||
Theorem | elrefrels2 36642 | Element of the class of reflexive relations. (Contributed by Peter Mazsa, 23-Jul-2019.) |
⊢ (𝑅 ∈ RefRels ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ 𝑅 ∈ Rels )) | ||
Theorem | elrefrels3 36643* | Element of the class of reflexive relations. (Contributed by Peter Mazsa, 23-Jul-2019.) |
⊢ (𝑅 ∈ RefRels ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) ∧ 𝑅 ∈ Rels )) | ||
Theorem | elrefrelsrel 36644 | For sets, being an element of the class of reflexive relations (df-refrels 36636) is equivalent to satisfying the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅)) | ||
Theorem | refreleq 36645 | Equality theorem for reflexive relation. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.) |
⊢ (𝑅 = 𝑆 → ( RefRel 𝑅 ↔ RefRel 𝑆)) | ||
Theorem | refrelid 36646 | Identity relation is reflexive. (Contributed by Peter Mazsa, 25-Jul-2021.) |
⊢ RefRel I | ||
Theorem | refrelcoss 36647 | The class of cosets by 𝑅 is reflexive. (Contributed by Peter Mazsa, 4-Jul-2020.) |
⊢ RefRel ≀ 𝑅 | ||
Definition | df-cnvrefs 36648 | Define the class of all converse reflexive sets, see the comment of df-ssr 36623. It is used only by df-cnvrefrels 36649. (Contributed by Peter Mazsa, 22-Jul-2019.) |
⊢ CnvRefs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥))◡ S (𝑥 ∩ (dom 𝑥 × ran 𝑥))} | ||
Definition | df-cnvrefrels 36649 |
Define the class of converse reflexive relations. This is practically
dfcnvrefrels2 36651 (which uses the traditional subclass
relation ⊆) :
we use converse subset relation (brcnvssr 36631) here to ensure the
comparability to the definitions of the classes of all reflexive
(df-ref 22665), symmetric (df-syms 36663) and transitive (df-trs 36693) sets.
We use this concept to define functions (df-funsALTV 36799, df-funALTV 36800) and disjoints (df-disjs 36822, df-disjALTV 36823). For sets, being an element of the class of converse reflexive relations is equivalent to satisfying the converse reflexive relation predicate, see elcnvrefrelsrel 36657. Alternate definitions are dfcnvrefrels2 36651 and dfcnvrefrels3 36652. (Contributed by Peter Mazsa, 7-Jul-2019.) |
⊢ CnvRefRels = ( CnvRefs ∩ Rels ) | ||
Definition | df-cnvrefrel 36650 | Define the converse reflexive relation predicate (read: 𝑅 is a converse reflexive relation), see also the comment of dfcnvrefrel3 36654. Alternate definitions are dfcnvrefrel2 36653 and dfcnvrefrel3 36654. (Contributed by Peter Mazsa, 16-Jul-2021.) |
⊢ ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)) | ||
Theorem | dfcnvrefrels2 36651 | Alternate definition of the class of converse reflexive relations. See the comment of dfrefrels2 36638. (Contributed by Peter Mazsa, 21-Jul-2021.) |
⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))} | ||
Theorem | dfcnvrefrels3 36652* | Alternate definition of the class of converse reflexive relations. (Contributed by Peter Mazsa, 22-Jul-2019.) |
⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)} | ||
Theorem | dfcnvrefrel2 36653 | Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 24-Jul-2019.) |
⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)) | ||
Theorem | dfcnvrefrel3 36654* | Alternate definition of the converse reflexive relation predicate. A relation is converse reflexive iff: for all elements on its domain and range, if for an element of its domain and for an element of its range there is the relation between them, then the two elements are the same, cf. the comment of dfrefrel3 36641. (Contributed by Peter Mazsa, 25-Jul-2021.) |
⊢ ( CnvRefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ Rel 𝑅)) | ||
Theorem | elcnvrefrels2 36655 | Element of the class of converse reflexive relations. (Contributed by Peter Mazsa, 25-Jul-2019.) |
⊢ (𝑅 ∈ CnvRefRels ↔ (𝑅 ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ 𝑅 ∈ Rels )) | ||
Theorem | elcnvrefrels3 36656* | Element of the class of converse reflexive relations. (Contributed by Peter Mazsa, 30-Aug-2021.) |
⊢ (𝑅 ∈ CnvRefRels ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ 𝑅 ∈ Rels )) | ||
Theorem | elcnvrefrelsrel 36657 | For sets, being an element of the class of converse reflexive relations (df-cnvrefrels 36649) is equivalent to satisfying the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅)) | ||
Theorem | cnvrefrelcoss2 36658 | Necessary and sufficient condition for a coset relation to be a converse reflexive relation. (Contributed by Peter Mazsa, 27-Jul-2021.) |
⊢ ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ I ) | ||
Theorem | cosselcnvrefrels2 36659 | Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 25-Aug-2021.) |
⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels )) | ||
Theorem | cosselcnvrefrels3 36660* | Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 30-Aug-2021.) |
⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ (∀𝑢∀𝑥∀𝑦((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝑅 ∈ Rels )) | ||
Theorem | cosselcnvrefrels4 36661* | Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 31-Aug-2021.) |
⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ (∀𝑢∃*𝑥 𝑢𝑅𝑥 ∧ ≀ 𝑅 ∈ Rels )) | ||
Theorem | cosselcnvrefrels5 36662* | Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 5-Sep-2021.) |
⊢ ( ≀ 𝑅 ∈ CnvRefRels ↔ (∀𝑥 ∈ ran 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]◡𝑅 ∩ [𝑦]◡𝑅) = ∅) ∧ ≀ 𝑅 ∈ Rels )) | ||
Definition | df-syms 36663 |
Define the class of all symmetric sets. It is used only by df-symrels 36664.
Note the similarity of Definitions df-refs 36635, df-syms 36663 and df-trs 36693, cf. the comment of dfrefrels2 36638. (Contributed by Peter Mazsa, 19-Jul-2019.) |
⊢ Syms = {𝑥 ∣ ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))} | ||
Definition | df-symrels 36664 |
Define the class of symmetric relations. For sets, being an element of
the class of symmetric relations is equivalent to satisfying the symmetric
relation predicate, see elsymrelsrel 36678. Alternate definitions are
dfsymrels2 36666, dfsymrels3 36667, dfsymrels4 36668 and dfsymrels5 36669.
This definition is similar to the definitions of the classes of reflexive (df-refrels 36636) and transitive (df-trrels 36694) relations. (Contributed by Peter Mazsa, 7-Jul-2019.) |
⊢ SymRels = ( Syms ∩ Rels ) | ||
Definition | df-symrel 36665 | Define the symmetric relation predicate. (Read: 𝑅 is a symmetric relation.) For sets, being an element of the class of symmetric relations (df-symrels 36664) is equivalent to satisfying the symmetric relation predicate, see elsymrelsrel 36678. Alternate definitions are dfsymrel2 36670 and dfsymrel3 36671. (Contributed by Peter Mazsa, 16-Jul-2021.) |
⊢ ( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)) | ||
Theorem | dfsymrels2 36666 | Alternate definition of the class of symmetric relations. Cf. the comment of dfrefrels2 36638. (Contributed by Peter Mazsa, 20-Jul-2019.) |
⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟} | ||
Theorem | dfsymrels3 36667* | Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.) |
⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥)} | ||
Theorem | dfsymrels4 36668 | Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 20-Jul-2019.) |
⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 = 𝑟} | ||
Theorem | dfsymrels5 36669* | Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.) |
⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} | ||
Theorem | dfsymrel2 36670 | Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 19-Apr-2019.) (Revised by Peter Mazsa, 17-Aug-2021.) |
⊢ ( SymRel 𝑅 ↔ (◡𝑅 ⊆ 𝑅 ∧ Rel 𝑅)) | ||
Theorem | dfsymrel3 36671* | Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 21-Apr-2019.) (Revised by Peter Mazsa, 17-Aug-2021.) |
⊢ ( SymRel 𝑅 ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ Rel 𝑅)) | ||
Theorem | dfsymrel4 36672 | Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.) |
⊢ ( SymRel 𝑅 ↔ (◡𝑅 = 𝑅 ∧ Rel 𝑅)) | ||
Theorem | dfsymrel5 36673* | Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.) |
⊢ ( SymRel 𝑅 ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ Rel 𝑅)) | ||
Theorem | elsymrels2 36674 | Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.) |
⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 ⊆ 𝑅 ∧ 𝑅 ∈ Rels )) | ||
Theorem | elsymrels3 36675* | Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.) |
⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels )) | ||
Theorem | elsymrels4 36676 | Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.) |
⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 = 𝑅 ∧ 𝑅 ∈ Rels )) | ||
Theorem | elsymrels5 36677* | Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.) |
⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels )) | ||
Theorem | elsymrelsrel 36678 | For sets, being an element of the class of symmetric relations (df-symrels 36664) is equivalent to satisfying the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅)) | ||
Theorem | symreleq 36679 | Equality theorem for symmetric relation. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.) |
⊢ (𝑅 = 𝑆 → ( SymRel 𝑅 ↔ SymRel 𝑆)) | ||
Theorem | symrelim 36680 | Symmetric relation implies that the domain and the range are equal. (Contributed by Peter Mazsa, 29-Dec-2021.) |
⊢ ( SymRel 𝑅 → dom 𝑅 = ran 𝑅) | ||
Theorem | symrelcoss 36681 | The class of cosets by 𝑅 is symmetric. (Contributed by Peter Mazsa, 20-Dec-2021.) |
⊢ SymRel ≀ 𝑅 | ||
Theorem | idsymrel 36682 | The identity relation is symmetric. (Contributed by AV, 19-Jun-2022.) |
⊢ SymRel I | ||
Theorem | epnsymrel 36683 | The membership (epsilon) relation is not symmetric. (Contributed by AV, 18-Jun-2022.) |
⊢ ¬ SymRel E | ||
Theorem | symrefref2 36684 | Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 36685. (Contributed by Peter Mazsa, 19-Jul-2018.) |
⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅)) | ||
Theorem | symrefref3 36685* | Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref2 36684. (Contributed by Peter Mazsa, 23-Aug-2021.) (Proof modification is discouraged.) |
⊢ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) → (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥)) | ||
Theorem | refsymrels2 36686 | Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels2 36708) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 36638, cf. the comment of dfrefrels2 36638. (Contributed by Peter Mazsa, 20-Jul-2019.) |
⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} | ||
Theorem | refsymrels3 36687* | Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels3 36709) can use the ∀𝑥 ∈ dom 𝑟𝑥𝑟𝑥 version for their reflexive part, not just the ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥 = 𝑦 → 𝑥𝑟𝑦) version of dfrefrels3 36639, cf. the comment of dfrefrel3 36641. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.) |
⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥))} | ||
Theorem | refsymrel2 36688 | A relation which is reflexive and symmetric (like an equivalence relation) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrel2 36640, cf. the comment of dfrefrels2 36638. (Contributed by Peter Mazsa, 23-Aug-2021.) |
⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅)) | ||
Theorem | refsymrel3 36689* | A relation which is reflexive and symmetric (like an equivalence relation) can use the ∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 version for its reflexive part, not just the ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) version of dfrefrel3 36641, cf. the comment of dfrefrel3 36641. (Contributed by Peter Mazsa, 23-Aug-2021.) |
⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ Rel 𝑅)) | ||
Theorem | elrefsymrels2 36690 | Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels2 36708) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrels2 36638, cf. the comment of dfrefrels2 36638. (Contributed by Peter Mazsa, 22-Jul-2019.) |
⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ 𝑅 ∈ Rels )) | ||
Theorem | elrefsymrels3 36691* | Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels3 36709) can use the ∀𝑥 ∈ dom 𝑅𝑥𝑅𝑥 version for their reflexive part, not just the ∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) version of dfrefrels3 36639, cf. the comment of dfrefrel3 36641. (Contributed by Peter Mazsa, 22-Jul-2019.) (Proof modification is discouraged.) |
⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ 𝑅 ∈ Rels )) | ||
Theorem | elrefsymrelsrel 36692 | For sets, being an element of the class of reflexive and symmetric relations is equivalent to satisfying the reflexive and symmetric relation predicates. (Contributed by Peter Mazsa, 23-Aug-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ( RefRel 𝑅 ∧ SymRel 𝑅))) | ||
Definition | df-trs 36693 |
Define the class of all transitive sets (versus the transitive class
defined in df-tr 5193). It is used only by df-trrels 36694.
Note the similarity of the definitions of df-refs 36635, df-syms 36663 and df-trs 36693. (Contributed by Peter Mazsa, 17-Jul-2021.) |
⊢ Trs = {𝑥 ∣ ((𝑥 ∩ (dom 𝑥 × ran 𝑥)) ∘ (𝑥 ∩ (dom 𝑥 × ran 𝑥))) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))} | ||
Definition | df-trrels 36694 |
Define the class of transitive relations. For sets, being an element of
the class of transitive relations is equivalent to satisfying the
transitive relation predicate, see eltrrelsrel 36702. Alternate definitions
are dftrrels2 36696 and dftrrels3 36697.
This definition is similar to the definitions of the classes of reflexive (df-refrels 36636) and symmetric (df-symrels 36664) relations. (Contributed by Peter Mazsa, 7-Jul-2019.) |
⊢ TrRels = ( Trs ∩ Rels ) | ||
Definition | df-trrel 36695 | Define the transitive relation predicate. (Read: 𝑅 is a transitive relation.) For sets, being an element of the class of transitive relations (df-trrels 36694) is equivalent to satisfying the transitive relation predicate, see eltrrelsrel 36702. Alternate definitions are dftrrel2 36698 and dftrrel3 36699. (Contributed by Peter Mazsa, 17-Jul-2021.) |
⊢ ( TrRel 𝑅 ↔ (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∘ (𝑅 ∩ (dom 𝑅 × ran 𝑅))) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)) | ||
Theorem | dftrrels2 36696 |
Alternate definition of the class of transitive relations.
I'd prefer to define the class of transitive relations by using the definition of composition by [Suppes] p. 63. df-coSUP (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥𝐴𝑢 ∧ 𝑢𝐵𝑦)} as opposed to the present definition of composition df-co 5599 (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥𝐵𝑢 ∧ 𝑢𝐴𝑦)} because the Suppes definition keeps the order of 𝐴, 𝐵, 𝐶, 𝑅, 𝑆, 𝑇 by default in trsinxpSUP (((𝑅 ∩ (𝐴 × 𝐵)) ∘ (𝑆 ∩ (𝐵 × 𝐶))) ⊆ (𝑇 ∩ (𝐴 × 𝐶)) ↔ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵∀ 𝑧 ∈ 𝐶((𝑥𝑅𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑇𝑧)) while the present definition of composition disarranges them: trsinxp (((𝑆 ∩ (𝐵 × 𝐶)) ∘ (𝑅 ∩ (𝐴 × 𝐵))) ⊆ (𝑇 ∩ (𝐴 × 𝐶 )) ↔ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐵∀𝑧 ∈ 𝐶((𝑥𝑅𝑦 ∧ 𝑦𝑆𝑧) → 𝑥𝑇𝑧) ). This is not mission critical to me, the implication of the Suppes definition is just more aesthetic, at least in the above case. If we swap to the Suppes definition of class composition, I would define the present class of all transitive sets as df-trsSUP and I would consider to switch the definition of the class of cosets by 𝑅 from the present df-coss 36544 to a df-cossSUP. But perhaps there is a mathematical reason to keep the present definition of composition. (Contributed by Peter Mazsa, 21-Jul-2021.) |
⊢ TrRels = {𝑟 ∈ Rels ∣ (𝑟 ∘ 𝑟) ⊆ 𝑟} | ||
Theorem | dftrrels3 36697* | Alternate definition of the class of transitive relations. (Contributed by Peter Mazsa, 22-Jul-2021.) |
⊢ TrRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)} | ||
Theorem | dftrrel2 36698 | Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.) |
⊢ ( TrRel 𝑅 ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅)) | ||
Theorem | dftrrel3 36699* | Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.) |
⊢ ( TrRel 𝑅 ↔ (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ Rel 𝑅)) | ||
Theorem | eltrrels2 36700 | Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.) |
⊢ (𝑅 ∈ TrRels ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ 𝑅 ∈ Rels )) |
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