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| Description: Define the coelement equivalence relations class, the class of sets with coelement equivalence relations. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 38597. Alternate definition is dfcoeleqvrels 38622. (Contributed by Peter Mazsa, 28-Nov-2022.) | 
| Ref | Expression | 
|---|---|
| df-coeleqvrels | ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ccoeleqvrels 38200 | . 2 class CoElEqvRels | |
| 2 | cep 5583 | . . . . . . 7 class E | |
| 3 | 2 | ccnv 5684 | . . . . . 6 class ◡ E | 
| 4 | va | . . . . . . 7 setvar 𝑎 | |
| 5 | 4 | cv 1539 | . . . . . 6 class 𝑎 | 
| 6 | 3, 5 | cres 5687 | . . . . 5 class (◡ E ↾ 𝑎) | 
| 7 | 6 | ccoss 38182 | . . . 4 class ≀ (◡ E ↾ 𝑎) | 
| 8 | ceqvrels 38198 | . . . 4 class EqvRels | |
| 9 | 7, 8 | wcel 2108 | . . 3 wff ≀ (◡ E ↾ 𝑎) ∈ EqvRels | 
| 10 | 9, 4 | cab 2714 | . 2 class {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } | 
| 11 | 1, 10 | wceq 1540 | 1 wff CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } | 
| Colors of variables: wff setvar class | 
| This definition is referenced by: elcoeleqvrels 38596 dfcoeleqvrels 38622 | 
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