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| Mirrors > Home > MPE Home > Th. List > Mathboxes > df-coeleqvrel | Structured version Visualization version GIF version | ||
| Description: Define the coelement equivalence relation predicate. (Read: the coelement equivalence relation on 𝐴.) Alternate definition is dfcoeleqvrel 38623. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 38597. (Contributed by Peter Mazsa, 11-Dec-2021.) | 
| Ref | Expression | 
|---|---|
| df-coeleqvrel | ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cA | . . 3 class 𝐴 | |
| 2 | 1 | wcoeleqvrel 38201 | . 2 wff CoElEqvRel 𝐴 | 
| 3 | cep 5583 | . . . . . 6 class E | |
| 4 | 3 | ccnv 5684 | . . . . 5 class ◡ E | 
| 5 | 4, 1 | cres 5687 | . . . 4 class (◡ E ↾ 𝐴) | 
| 6 | 5 | ccoss 38182 | . . 3 class ≀ (◡ E ↾ 𝐴) | 
| 7 | 6 | weqvrel 38199 | . 2 wff EqvRel ≀ (◡ E ↾ 𝐴) | 
| 8 | 2, 7 | wb 206 | 1 wff ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) | 
| Colors of variables: wff setvar class | 
| This definition is referenced by: elcoeleqvrelsrel 38597 dfcoeleqvrel 38623 eqvreldmqs 38676 eldisjim 38785 | 
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