Definition df-ers 39247

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 8678 "is not standard", "somewhat cryptic", has no constant 0-ary class and does not follow the traditional transparent reflexive-symmetric-transitive relation way of definition of equivalence. Definitions df-eqvrels 39167, dfeqvrels2 39171, dfeqvrels3 39172 and df-eqvrel 39168, dfeqvrel2 39173, dfeqvrel3 39174 are fully transparent in this regard. However, they lack the domain component (dom 𝑅 = 𝐴) of the present df-er 8678. While we acknowledge the need of a domain component, the present df-er 8678 definition does not utilize the results revealed by the new theorems in the Partition-Equivalence Theorem part below (like pets 39465 and pet 39464). From those theorems follows that the natural domain of equivalence relations is

not 𝑅Domain𝐴 (i.e. dom 𝑅 = 𝐴 see brdomaing 36283),

but 𝑅 DomainQss 𝐴 (i.e. (dom 𝑅 / 𝑅) = 𝐴, see brdmqss 39229), see erimeq 39263 vs. prter3 39506.

While I'm sure we need both equivalence relation df-eqvrels 39167 and equivalence relation on domain quotient df-ers 39247, 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 39167 and df-ers 39247 are enough and named the predicate version of the one on domain quotient as the alternate version df-erALTV 39248 of the present df-er 8678. (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 38707	. 2 class Ers
2		cdmqss 38705	. . 3 class DomainQss
3		ceqvrels 38698	. . 3 class EqvRels
4	2, 3	cres 5649	. 2 class ( DomainQss ↾ EqvRels )
5	1, 4	wceq 1560	1 wff Ers = ( DomainQss ↾ EqvRels )

This definition is referenced by:  brers  39251