Definition df-coeleqvrel 38578

Description: Define the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.) Alternate definition is dfcoeleqvrel 38613. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 38587. (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 38188	. 2 wff CoElEqvRel 𝐴
3		cep 5537	. . . . . 6 class E
4	3	ccnv 5637	. . . . 5 class ◡ E
5	4, 1	cres 5640	. . . 4 class (◡ E ↾ 𝐴)
6	5	ccoss 38169	. . 3 class ≀ (◡ E ↾ 𝐴)
7	6	weqvrel 38186	. 2 wff EqvRel ≀ (◡ E ↾ 𝐴)
8	2, 7	wb 206	1 wff ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))

This definition is referenced by:  elcoeleqvrelsrel  38587  dfcoeleqvrel  38613  eqvreldmqs  38667  eldisjim  38776