Definition df-coeleqvrel 38569

Description: Define the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.) Alternate definition is dfcoeleqvrel 38604. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 38578. (Contributed by Peter Mazsa, 11-Dec-2021.)

Assertion

Ref	Expression
df-coeleqvrel	⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))

Detailed syntax breakdown of Definition df-coeleqvrel

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	wcoeleqvrel 38181	. 2 wff CoElEqvRel 𝐴
3		cep 5588	. . . . . 6 class E
4	3	ccnv 5688	. . . . 5 class ◡ E
5	4, 1	cres 5691	. . . 4 class (◡ E ↾ 𝐴)
6	5	ccoss 38162	. . 3 class ≀ (◡ E ↾ 𝐴)
7	6	weqvrel 38179	. 2 wff EqvRel ≀ (◡ E ↾ 𝐴)
8	2, 7	wb 206	1 wff ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))

This definition is referenced by:  elcoeleqvrelsrel  38578  dfcoeleqvrel  38604  eqvreldmqs  38657  eldisjim  38766