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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | brssr 35901 | The subset relation and subclass relationship (df-ss 3898) 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 35902 | Any set is a subset of itself. (Contributed by Peter Mazsa, 1-Aug-2019.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 S 𝐴) | ||
Theorem | issetssr 35903 | Two ways of expressing set existence. (Contributed by Peter Mazsa, 1-Aug-2019.) |
⊢ (𝐴 ∈ V ↔ 𝐴 S 𝐴) | ||
Theorem | brssrres 35904 | Restricted subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.) |
⊢ (𝐶 ∈ 𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ⊆ 𝐶))) | ||
Theorem | br1cnvssrres 35905 | Restricted converse subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.) |
⊢ (𝐵 ∈ 𝑉 → (𝐵◡( S ↾ 𝐴)𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐵))) | ||
Theorem | brcnvssr 35906 | The converse of a subset relation swaps arguments. (Contributed by Peter Mazsa, 1-Aug-2019.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴◡ S 𝐵 ↔ 𝐵 ⊆ 𝐴)) | ||
Theorem | brcnvssrid 35907 | Any set is a converse subset of itself. (Contributed by Peter Mazsa, 9-Jun-2021.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴◡ S 𝐴) | ||
Theorem | br1cossxrncnvssrres 35908* | 〈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 35909 | Property of subset relation, see also extid 35728, extep 35700 and the comment of df-ssr 35898. (Contributed by Peter Mazsa, 10-Jul-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴]◡ S = [𝐵]◡ S ↔ 𝐴 = 𝐵)) | ||
Definition | df-refs 35910 |
Define the class of all reflexive sets. It is used only by df-refrels 35911.
We use subset relation S (df-ssr 35898) here to be able to define
converse reflexivity (df-cnvrefs 35923), see also the comment of df-ssr 35898.
The elements of this class are not necessarily relations (versus
df-refrels 35911).
Note the similarity of the definitions df-refs 35910, df-syms 35938 and df-trs 35968, cf. comments of dfrefrels2 35913. (Contributed by Peter Mazsa, 19-Jul-2019.) |
⊢ Refs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))} | ||
Definition | df-refrels 35911 |
Define the class of reflexive relations. This is practically dfrefrels2 35913
(which reveals that RefRels can not include proper
classes like I
as is elements, see comments of dfrefrels2 35913).
Another alternative definition is dfrefrels3 35914. The element of this class and the reflexive relation predicate (df-refrel 35912) are the same, that is, (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝐴 is a set, see elrefrelsrel 35919. This definition is similar to the definitions of the classes of symmetric (df-symrels 35939) and transitive (df-trrels 35969) relations. (Contributed by Peter Mazsa, 7-Jul-2019.) |
⊢ RefRels = ( Refs ∩ Rels ) | ||
Definition | df-refrel 35912 | Define the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.) This is a surprising definition, see the comment of dfrefrel3 35916. Alternate definitions are dfrefrel2 35915 and dfrefrel3 35916. For sets, being an element of the class of reflexive relations (df-refrels 35911) is equivalent to satisfying the reflexive relation predicate, that is (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝑅 is a set, see elrefrelsrel 35919. (Contributed by Peter Mazsa, 16-Jul-2021.) |
⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)) | ||
Theorem | dfrefrels2 35913 |
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 7600)
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 3459. 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 35912. See
also the comment of df-rels 35885.
Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 35913, it keeps restriction of I: this is why the very similar definitions df-refs 35910, df-syms 35938 and df-trs 35968 diverge when we switch from (general) sets to relations in dfrefrels2 35913, dfsymrels2 35941 and dftrrels2 35971. (Contributed by Peter Mazsa, 20-Jul-2019.) |
⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} | ||
Theorem | dfrefrels3 35914* | 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 35915 | 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 35916* |
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 6885 / idrefALT 5940 or df-reflexive 45294 ⊢ (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 35962. Only when this symmetry condition holds, like in case of equivalence relations, see dfeqvrels3 35984, 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 35737 where ⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 = 𝑦 → 𝑥𝑅𝑦) ↔ ∀𝑥 ∈ 𝐴𝑥𝑅𝑥). See also similar definition of the converse reflexive relations class dfcnvrefrel3 35929. (Contributed by Peter Mazsa, 8-Jul-2019.) |
⊢ ( RefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥 = 𝑦 → 𝑥𝑅𝑦) ∧ Rel 𝑅)) | ||
Theorem | elrefrels2 35917 | 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 35918* | 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 35919 | For sets, being an element of the class of reflexive relations (df-refrels 35911) is equivalent to satisfying the reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ RefRels ↔ RefRel 𝑅)) | ||
Theorem | refreleq 35920 | 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 35921 | Identity relation is reflexive. (Contributed by Peter Mazsa, 25-Jul-2021.) |
⊢ RefRel I | ||
Theorem | refrelcoss 35922 | The class of cosets by 𝑅 is reflexive. (Contributed by Peter Mazsa, 4-Jul-2020.) |
⊢ RefRel ≀ 𝑅 | ||
Definition | df-cnvrefs 35923 | Define the class of all converse reflexive sets, see the comment of df-ssr 35898. It is used only by df-cnvrefrels 35924. (Contributed by Peter Mazsa, 22-Jul-2019.) |
⊢ CnvRefs = {𝑥 ∣ ( I ∩ (dom 𝑥 × ran 𝑥))◡ S (𝑥 ∩ (dom 𝑥 × ran 𝑥))} | ||
Definition | df-cnvrefrels 35924 |
Define the class of converse reflexive relations. This is practically
dfcnvrefrels2 35926 (which uses the traditional subclass
relation ⊆) :
we use converse subset relation (brcnvssr 35906) here to ensure the
comparability to the definitions of the classes of all reflexive
(df-ref 22110), symmetric (df-syms 35938) and transitive (df-trs 35968) sets.
We use this concept to define functions (df-funsALTV 36074, df-funALTV 36075) and disjoints (df-disjs 36097, df-disjALTV 36098). For sets, being an element of the class of converse reflexive relations is equivalent to satisfying the converse reflexive relation predicate, see elcnvrefrelsrel 35932. Alternate definitions are dfcnvrefrels2 35926 and dfcnvrefrels3 35927. (Contributed by Peter Mazsa, 7-Jul-2019.) |
⊢ CnvRefRels = ( CnvRefs ∩ Rels ) | ||
Definition | df-cnvrefrel 35925 | Define the converse reflexive relation predicate (read: 𝑅 is a converse reflexive relation), see also the comment of dfcnvrefrel3 35929. Alternate definitions are dfcnvrefrel2 35928 and dfcnvrefrel3 35929. (Contributed by Peter Mazsa, 16-Jul-2021.) |
⊢ ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)) | ||
Theorem | dfcnvrefrels2 35926 | Alternate definition of the class of converse reflexive relations. See the comment of dfrefrels2 35913. (Contributed by Peter Mazsa, 21-Jul-2021.) |
⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))} | ||
Theorem | dfcnvrefrels3 35927* | 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 35928 | 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 35929* | 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 35916. (Contributed by Peter Mazsa, 25-Jul-2021.) |
⊢ ( CnvRefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅∀𝑦 ∈ ran 𝑅(𝑥𝑅𝑦 → 𝑥 = 𝑦) ∧ Rel 𝑅)) | ||
Theorem | elcnvrefrels2 35930 | 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 35931* | 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 35932 | For sets, being an element of the class of converse reflexive relations (df-cnvrefrels 35924) is equivalent to satisfying the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-Jul-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ CnvRefRels ↔ CnvRefRel 𝑅)) | ||
Theorem | cnvrefrelcoss2 35933 | 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 35934 | 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 35935* | 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 35936* | 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 35937* | 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 35938 |
Define the class of all symmetric sets. It is used only by df-symrels 35939.
Note the similarity of the definitions df-refs 35910, df-syms 35938 and df-trs 35968, cf. the comment of dfrefrels2 35913. (Contributed by Peter Mazsa, 19-Jul-2019.) |
⊢ Syms = {𝑥 ∣ ◡(𝑥 ∩ (dom 𝑥 × ran 𝑥)) S (𝑥 ∩ (dom 𝑥 × ran 𝑥))} | ||
Definition | df-symrels 35939 |
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 35953. Alternate definitions are
dfsymrels2 35941, dfsymrels3 35942, dfsymrels4 35943 and dfsymrels5 35944.
This definition is similar to the definitions of the classes of reflexive (df-refrels 35911) and transitive (df-trrels 35969) relations. (Contributed by Peter Mazsa, 7-Jul-2019.) |
⊢ SymRels = ( Syms ∩ Rels ) | ||
Definition | df-symrel 35940 | Define the symmetric relation predicate. (Read: 𝑅 is a symmetric relation.) For sets, being an element of the class of symmetric relations (df-symrels 35939) is equivalent to satisfying the symmetric relation predicate, see elsymrelsrel 35953. Alternate definitions are dfsymrel2 35945 and dfsymrel3 35946. (Contributed by Peter Mazsa, 16-Jul-2021.) |
⊢ ( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)) | ||
Theorem | dfsymrels2 35941 | Alternate definition of the class of symmetric relations. Cf. the comment of dfrefrels2 35913. (Contributed by Peter Mazsa, 20-Jul-2019.) |
⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟} | ||
Theorem | dfsymrels3 35942* | Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.) |
⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥)} | ||
Theorem | dfsymrels4 35943 | Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 20-Jul-2019.) |
⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 = 𝑟} | ||
Theorem | dfsymrels5 35944* | Alternate definition of the class of symmetric relations. (Contributed by Peter Mazsa, 22-Jul-2021.) |
⊢ SymRels = {𝑟 ∈ Rels ∣ ∀𝑥∀𝑦(𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} | ||
Theorem | dfsymrel2 35945 | 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 35946* | 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 35947 | Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.) |
⊢ ( SymRel 𝑅 ↔ (◡𝑅 = 𝑅 ∧ Rel 𝑅)) | ||
Theorem | dfsymrel5 35948* | Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.) |
⊢ ( SymRel 𝑅 ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ Rel 𝑅)) | ||
Theorem | elsymrels2 35949 | Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.) |
⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 ⊆ 𝑅 ∧ 𝑅 ∈ Rels )) | ||
Theorem | elsymrels3 35950* | Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.) |
⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels )) | ||
Theorem | elsymrels4 35951 | Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.) |
⊢ (𝑅 ∈ SymRels ↔ (◡𝑅 = 𝑅 ∧ 𝑅 ∈ Rels )) | ||
Theorem | elsymrels5 35952* | Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.) |
⊢ (𝑅 ∈ SymRels ↔ (∀𝑥∀𝑦(𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥) ∧ 𝑅 ∈ Rels )) | ||
Theorem | elsymrelsrel 35953 | For sets, being an element of the class of symmetric relations (df-symrels 35939) is equivalent to satisfying the symmetric relation predicate. (Contributed by Peter Mazsa, 17-Aug-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ SymRels ↔ SymRel 𝑅)) | ||
Theorem | symreleq 35954 | 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 35955 | 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 35956 | The class of cosets by 𝑅 is symmetric. (Contributed by Peter Mazsa, 20-Dec-2021.) |
⊢ SymRel ≀ 𝑅 | ||
Theorem | idsymrel 35957 | The identity relation is symmetric. (Contributed by AV, 19-Jun-2022.) |
⊢ SymRel I | ||
Theorem | epnsymrel 35958 | The membership (epsilon) relation is not symmetric. (Contributed by AV, 18-Jun-2022.) |
⊢ ¬ SymRel E | ||
Theorem | symrefref2 35959 | Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref3 35960. (Contributed by Peter Mazsa, 19-Jul-2018.) |
⊢ (◡𝑅 ⊆ 𝑅 → (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ( I ↾ dom 𝑅) ⊆ 𝑅)) | ||
Theorem | symrefref3 35960* | Symmetry is a sufficient condition for the equivalence of two versions of the reflexive relation, see also symrefref2 35959. (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 35961 | 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 35983) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟 version of dfrefrels2 35913, cf. the comment of dfrefrels2 35913. (Contributed by Peter Mazsa, 20-Jul-2019.) |
⊢ ( RefRels ∩ SymRels ) = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟)} | ||
Theorem | refsymrels3 35962* | 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 35984) 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 35914, cf. the comment of dfrefrel3 35916. (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 35963 | 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 35915, cf. the comment of dfrefrels2 35913. (Contributed by Peter Mazsa, 23-Aug-2021.) |
⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ Rel 𝑅)) | ||
Theorem | refsymrel3 35964* | 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 35916, cf. the comment of dfrefrel3 35916. (Contributed by Peter Mazsa, 23-Aug-2021.) |
⊢ (( RefRel 𝑅 ∧ SymRel 𝑅) ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥)) ∧ Rel 𝑅)) | ||
Theorem | elrefsymrels2 35965 | 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 35983) can use the restricted version for their reflexive part (see below), not just the ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 version of dfrefrels2 35913, cf. the comment of dfrefrels2 35913. (Contributed by Peter Mazsa, 22-Jul-2019.) |
⊢ (𝑅 ∈ ( RefRels ∩ SymRels ) ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅) ∧ 𝑅 ∈ Rels )) | ||
Theorem | elrefsymrels3 35966* | 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 35984) 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 35914, cf. the comment of dfrefrel3 35916. (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 35967 | 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 35968 |
Define the class of all transitive sets (versus the transitive class
defined in df-tr 5137). It is used only by df-trrels 35969.
Note the similarity of the definitions of df-refs 35910, df-syms 35938 and df-trs 35968. (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 35969 |
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 35977. Alternate definitions
are dftrrels2 35971 and dftrrels3 35972.
This definition is similar to the definitions of the classes of reflexive (df-refrels 35911) and symmetric (df-symrels 35939) relations. (Contributed by Peter Mazsa, 7-Jul-2019.) |
⊢ TrRels = ( Trs ∩ Rels ) | ||
Definition | df-trrel 35970 | Define the transitive relation predicate. (Read: 𝑅 is a transitive relation.) For sets, being an element of the class of transitive relations (df-trrels 35969) is equivalent to satisfying the transitive relation predicate, see eltrrelsrel 35977. Alternate definitions are dftrrel2 35973 and dftrrel3 35974. (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 35971 |
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 5528 (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 35819 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 35972* | 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 35973 | Alternate definition of the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.) |
⊢ ( TrRel 𝑅 ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ Rel 𝑅)) | ||
Theorem | dftrrel3 35974* | 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 35975 | Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.) |
⊢ (𝑅 ∈ TrRels ↔ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ 𝑅 ∈ Rels )) | ||
Theorem | eltrrels3 35976* | Element of the class of transitive relations. (Contributed by Peter Mazsa, 22-Aug-2021.) |
⊢ (𝑅 ∈ TrRels ↔ (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ 𝑅 ∈ Rels )) | ||
Theorem | eltrrelsrel 35977 | For sets, being an element of the class of transitive relations is equivalent to satisfying the transitive relation predicate. (Contributed by Peter Mazsa, 22-Aug-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ TrRels ↔ TrRel 𝑅)) | ||
Theorem | trreleq 35978 | Equality theorem for the transitive relation predicate. (Contributed by Peter Mazsa, 15-Apr-2019.) (Revised by Peter Mazsa, 23-Sep-2021.) |
⊢ (𝑅 = 𝑆 → ( TrRel 𝑅 ↔ TrRel 𝑆)) | ||
Definition | df-eqvrels 35979 | Define the class of equivalence relations. For sets, being an element of the class of equivalence relations is equivalent to satisfying the equivalence relation predicate, see eleqvrelsrel 35989. Alternate definitions are dfeqvrels2 35983 and dfeqvrels3 35984. (Contributed by Peter Mazsa, 7-Nov-2018.) |
⊢ EqvRels = (( RefRels ∩ SymRels ) ∩ TrRels ) | ||
Definition | df-eqvrel 35980 | Define the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.) For sets, being an element of the class of equivalence relations (df-eqvrels 35979) is equivalent to satisfying the equivalence relation predicate, see eleqvrelsrel 35989. Alternate definitions are dfeqvrel2 35985 and dfeqvrel3 35986. (Contributed by Peter Mazsa, 17-Apr-2019.) |
⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅)) | ||
Definition | df-coeleqvrels 35981 | Define the the coelement equivalence relations class, the class of sets with coelement equivalence relations. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 35991. Alternate definition is dfcoeleqvrels 36016. (Contributed by Peter Mazsa, 28-Nov-2022.) |
⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } | ||
Definition | df-coeleqvrel 35982 | Define the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.) Alternate definition is dfcoeleqvrel 36017. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 35991. (Contributed by Peter Mazsa, 11-Dec-2021.) |
⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) | ||
Theorem | dfeqvrels2 35983 | Alternate definition of the class of equivalence relations. (Contributed by Peter Mazsa, 2-Dec-2019.) |
⊢ EqvRels = {𝑟 ∈ Rels ∣ (( I ↾ dom 𝑟) ⊆ 𝑟 ∧ ◡𝑟 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} | ||
Theorem | dfeqvrels3 35984* | Alternate definition of the class of equivalence relations. (Contributed by Peter Mazsa, 2-Dec-2019.) |
⊢ EqvRels = {𝑟 ∈ Rels ∣ (∀𝑥 ∈ dom 𝑟 𝑥𝑟𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑟𝑦 → 𝑦𝑟𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))} | ||
Theorem | dfeqvrel2 35985 | Alternate definition of the equivalence relation predicate. (Contributed by Peter Mazsa, 22-Apr-2019.) |
⊢ ( EqvRel 𝑅 ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ Rel 𝑅)) | ||
Theorem | dfeqvrel3 35986* | Alternate definition of the equivalence relation predicate. (Contributed by Peter Mazsa, 22-Apr-2019.) |
⊢ ( EqvRel 𝑅 ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ Rel 𝑅)) | ||
Theorem | eleqvrels2 35987 | Element of the class of equivalence relations. (Contributed by Peter Mazsa, 24-Aug-2021.) |
⊢ (𝑅 ∈ EqvRels ↔ ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ ◡𝑅 ⊆ 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) ∧ 𝑅 ∈ Rels )) | ||
Theorem | eleqvrels3 35988* | Element of the class of equivalence relations. (Contributed by Peter Mazsa, 24-Aug-2021.) |
⊢ (𝑅 ∈ EqvRels ↔ ((∀𝑥 ∈ dom 𝑅 𝑥𝑅𝑥 ∧ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ 𝑅 ∈ Rels )) | ||
Theorem | eleqvrelsrel 35989 | For sets, being an element of the class of equivalence relations is equivalent to satisfying the equivalence relation predicate. (Contributed by Peter Mazsa, 24-Aug-2021.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ EqvRels ↔ EqvRel 𝑅)) | ||
Theorem | elcoeleqvrels 35990 | Elementhood in the coelement equivalence relations class. (Contributed by Peter Mazsa, 24-Jul-2023.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels )) | ||
Theorem | elcoeleqvrelsrel 35991 | For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate. (Contributed by Peter Mazsa, 24-Jul-2023.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ CoElEqvRel 𝐴)) | ||
Theorem | eqvrelrel 35992 | An equivalence relation is a relation. (Contributed by Peter Mazsa, 2-Jun-2019.) |
⊢ ( EqvRel 𝑅 → Rel 𝑅) | ||
Theorem | eqvrelrefrel 35993 | An equivalence relation is reflexive. (Contributed by Peter Mazsa, 29-Dec-2021.) |
⊢ ( EqvRel 𝑅 → RefRel 𝑅) | ||
Theorem | eqvrelsymrel 35994 | An equivalence relation is symmetric. (Contributed by Peter Mazsa, 29-Dec-2021.) |
⊢ ( EqvRel 𝑅 → SymRel 𝑅) | ||
Theorem | eqvreltrrel 35995 | An equivalence relation is transitive. (Contributed by Peter Mazsa, 29-Dec-2021.) |
⊢ ( EqvRel 𝑅 → TrRel 𝑅) | ||
Theorem | eqvrelim 35996 | Equivalence relation implies that the domain and the range are equal. (Contributed by Peter Mazsa, 29-Dec-2021.) |
⊢ ( EqvRel 𝑅 → dom 𝑅 = ran 𝑅) | ||
Theorem | eqvreleq 35997 | Equality theorem for equivalence relation. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 23-Sep-2021.) |
⊢ (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆)) | ||
Theorem | eqvreleqi 35998 | Equality theorem for equivalence relation, inference version. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ 𝑅 = 𝑆 ⇒ ⊢ ( EqvRel 𝑅 ↔ EqvRel 𝑆) | ||
Theorem | eqvreleqd 35999 | Equality theorem for equivalence relation, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.) |
⊢ (𝜑 → 𝑅 = 𝑆) ⇒ ⊢ (𝜑 → ( EqvRel 𝑅 ↔ EqvRel 𝑆)) | ||
Theorem | eqvrelsym 36000 | An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 2-Jun-2019.) |
⊢ (𝜑 → EqvRel 𝑅) & ⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐵𝑅𝐴) |
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