Description: Define the class of
equivalence relations on domain quotients (or: domain
quotients restricted to equivalence relations).
The present definition of equivalence relation in set.mm df-er 8282 "is not
standard", "somewhat cryptic", has no costant 0-ary class
and does not
follow the traditional transparent reflexive-symmetric-transitive relation
way of definition of equivalence. The definitions df-eqvrels 35859,
dfeqvrels2 35863, dfeqvrels3 35864 and df-eqvrel 35860, dfeqvrel2 35865, dfeqvrel3 35866
are fully transparent in this regard. However, they lack the domain
component (dom 𝑅 = 𝐴) of the present df-er 8282. While we acknowledge
the need of a domain component, the present df-er 8282 definition does not
utilize the results revealed by the new theorems in the
Partition-Equivalence Theorem part below (like ~? pets and ~? pet ).
From those theorems follows that the natural domain of equivalence
relations is
not 𝑅Domain𝐴 (i.e. dom 𝑅 = 𝐴 see brdomaing 33415),
but 𝑅
DomainQss 𝐴 (i.e.
(dom 𝑅
/ 𝑅) = 𝐴, see brdmqss 35921), see
erim 35952 vs. prter3 36058.
While I'm sure we need both equivalence relation df-eqvrels 35859 and
equivalence relation on domain quotient df-ers 35937, I'm not sure whether we
need a third equivalence relation concept with the present dom 𝑅 = 𝐴
component as well: this needs further investigation. As a default I
suppose that these two concepts df-eqvrels 35859 and df-ers 35937 are enough and
named the predicate version of the one on domain quotient as the alternate
version df-erALTV 35938 of the present df-er 8282. (Contributed by Peter Mazsa,
26-Jun-2021.) |