Definition df-ers 36702

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 8456 "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. Definitions df-eqvrels 36624, dfeqvrels2 36628, dfeqvrels3 36629 and df-eqvrel 36625, dfeqvrel2 36630, dfeqvrel3 36631 are fully transparent in this regard. However, they lack the domain component (dom 𝑅 = 𝐴) of the present df-er 8456. While we acknowledge the need of a domain component, the present df-er 8456 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 34164),

but 𝑅 DomainQss 𝐴 (i.e. (dom 𝑅 / 𝑅) = 𝐴, see brdmqss 36686), see erim 36717 vs. prter3 36823.

While I'm sure we need both equivalence relation df-eqvrels 36624 and equivalence relation on domain quotient df-ers 36702, 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 36624 and df-ers 36702 are enough and named the predicate version of the one on domain quotient as the alternate version df-erALTV 36703 of the present df-er 8456. (Contributed by Peter Mazsa, 26-Jun-2021.)

Assertion

Ref	Expression
df-ers	⊢ Ers = ( DomainQss ↾ EqvRels )

Detailed syntax breakdown of Definition df-ers

Step	Hyp	Ref	Expression
1		cers 36285	. 2 class Ers
2		cdmqss 36283	. . 3 class DomainQss
3		ceqvrels 36276	. . 3 class EqvRels
4	2, 3	cres 5582	. 2 class ( DomainQss ↾ EqvRels )
5	1, 4	wceq 1539	1 wff Ers = ( DomainQss ↾ EqvRels )

This definition is referenced by:  brers  36706