Definition df-coeleqvrel 38543

Description: Define the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.) Alternate definition is dfcoeleqvrel 38578. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 38552. (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 38154	. 2 wff CoElEqvRel 𝐴
3		cep 5598	. . . . . 6 class E
4	3	ccnv 5699	. . . . 5 class ◡ E
5	4, 1	cres 5702	. . . 4 class (◡ E ↾ 𝐴)
6	5	ccoss 38135	. . 3 class ≀ (◡ E ↾ 𝐴)
7	6	weqvrel 38152	. 2 wff EqvRel ≀ (◡ E ↾ 𝐴)
8	2, 7	wb 206	1 wff ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))

This definition is referenced by:  elcoeleqvrelsrel  38552  dfcoeleqvrel  38578  eqvreldmqs  38631  eldisjim  38740