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Mirrors > Home > MPE Home > Th. List > Mathboxes > dferALTV2 | Structured version Visualization version GIF version |
Description: Equivalence relation with natural domain predicate, see the comment of df-ers 38644. (Contributed by Peter Mazsa, 26-Jun-2021.) (Revised by Peter Mazsa, 30-Aug-2021.) |
Ref | Expression |
---|---|
dferALTV2 | ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-erALTV 38645 | . 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴)) | |
2 | df-dmqs 38620 | . . 3 ⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴) | |
3 | 2 | anbi2i 623 | . 2 ⊢ (( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴) ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) |
4 | 1, 3 | bitri 275 | 1 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 dom cdm 5688 / cqs 8742 EqvRel weqvrel 38178 DomainQs wdmqs 38185 ErALTV werALTV 38187 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 207 df-an 396 df-dmqs 38620 df-erALTV 38645 |
This theorem is referenced by: erALTVeq1 38650 dfcomember2 38654 erimeq 38660 partim 38789 pet0 38796 petid 38798 petidres 38800 petinidres 38802 petxrnidres 38804 mainer 38815 petincnvepres 38830 pet 38832 |
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