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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 37113. For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate, see elcoeleqvrelsrel 37087. (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 36682 | . 2 wff CoElEqvRel 𝐴 |
3 | cep 5541 | . . . . . 6 class E | |
4 | 3 | ccnv 5637 | . . . . 5 class ◡ E |
5 | 4, 1 | cres 5640 | . . . 4 class (◡ E ↾ 𝐴) |
6 | 5 | ccoss 36663 | . . 3 class ≀ (◡ E ↾ 𝐴) |
7 | 6 | weqvrel 36680 | . 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 37087 dfcoeleqvrel 37113 eqvreldmqs 37166 eldisjim 37275 |
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