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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-erALTV | Structured version Visualization version GIF version |
Description: Equivalence relation with natural domain predicate, see also the comment of df-ers 36754. Alternate definition is dferALTV2 36759. Binary equivalence relation with natural domain and the equivalence relation with natural domain predicate are the same when 𝐴 and 𝑅 are sets, see brerser 36767. (Contributed by Peter Mazsa, 12-Aug-2021.) |
Ref | Expression |
---|---|
df-erALTV | ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cR | . . 3 class 𝑅 | |
3 | 1, 2 | werALTV 36338 | . 2 wff 𝑅 ErALTV 𝐴 |
4 | 2 | weqvrel 36329 | . . 3 wff EqvRel 𝑅 |
5 | 1, 2 | wdmqs 36336 | . . 3 wff 𝑅 DomainQs 𝐴 |
6 | 4, 5 | wa 395 | . 2 wff ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴) |
7 | 3, 6 | wb 205 | 1 wff (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴)) |
Colors of variables: wff setvar class |
This definition is referenced by: dferALTV2 36759 brerser 36767 |
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