Definition df-coeleqvrel 37549

Description: Define the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.) Alternate definition is dfcoeleqvrel 37584. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 37558. (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 37154	. 2 wff CoElEqvRel 𝐴
3		cep 5579	. . . . . 6 class E
4	3	ccnv 5675	. . . . 5 class ◡ E
5	4, 1	cres 5678	. . . 4 class (◡ E ↾ 𝐴)
6	5	ccoss 37135	. . 3 class ≀ (◡ E ↾ 𝐴)
7	6	weqvrel 37152	. 2 wff EqvRel ≀ (◡ E ↾ 𝐴)
8	2, 7	wb 205	1 wff ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))

This definition is referenced by:  elcoeleqvrelsrel  37558  dfcoeleqvrel  37584  eqvreldmqs  37637  eldisjim  37746