Definition df-ers 39680

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 8717 "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 39600, dfeqvrels2 39604, dfeqvrels3 39605 and df-eqvrel 39601, dfeqvrel2 39606, dfeqvrel3 39607 are fully transparent in this regard. However, they lack the domain component (dom 𝑅 = 𝐴) of the present df-er 8717. While we acknowledge the need of a domain component, the present df-er 8717 definition does not utilize the results revealed by the new theorems in the Partition-Equivalence Theorem part below (like pets 39898 and pet 39897). From those theorems follows that the natural domain of equivalence relations is

not 𝑅Domain𝐴 (i.e. dom 𝑅 = 𝐴 see brdomaing 36697),

but 𝑅 DomainQss 𝐴 (i.e. (dom 𝑅 / 𝑅) = 𝐴, see brdmqss 39662), see erimeq 39696 vs. prter3 39939.

While I'm sure we need both equivalence relation df-eqvrels 39600 and equivalence relation on domain quotient df-ers 39680, 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 39600 and df-ers 39680 are enough and named the predicate version of the one on domain quotient as the alternate version df-erALTV 39681 of the present df-er 8717. (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 39140	. 2 class Ers
2		cdmqss 39138	. . 3 class DomainQss
3		ceqvrels 39131	. . 3 class EqvRels
4	2, 3	cres 5653	. 2 class ( DomainQss ↾ EqvRels )
5	1, 4	wceq 1570	1 wff Ers = ( DomainQss ↾ EqvRels )

This definition is used by:  brers  39684