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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 38664. (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 38665 | . 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴)) | |
| 2 | df-dmqs 38640 | . . 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 1540 dom cdm 5685 / cqs 8744 EqvRel weqvrel 38199 DomainQs wdmqs 38206 ErALTV werALTV 38208 |
| 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 38640 df-erALTV 38665 |
| This theorem is referenced by: erALTVeq1 38670 dfcomember2 38674 erimeq 38680 partim 38809 pet0 38816 petid 38818 petidres 38820 petinidres 38822 petxrnidres 38824 mainer 38835 petincnvepres 38850 pet 38852 |
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