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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 37430. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 37404. (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 37000 | . 2 wff CoElEqvRel 𝐴 |
3 | cep 5578 | . . . . . 6 class E | |
4 | 3 | ccnv 5674 | . . . . 5 class ◡ E |
5 | 4, 1 | cres 5677 | . . . 4 class (◡ E ↾ 𝐴) |
6 | 5 | ccoss 36981 | . . 3 class ≀ (◡ E ↾ 𝐴) |
7 | 6 | weqvrel 36998 | . 2 wff EqvRel ≀ (◡ E ↾ 𝐴) |
8 | 2, 7 | wb 205 | 1 wff ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) |
Colors of variables: wff setvar class |
This definition is referenced by: elcoeleqvrelsrel 37404 dfcoeleqvrel 37430 eqvreldmqs 37483 eldisjim 37592 |
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