Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
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 36754. (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 36755 | . 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴)) | |
2 | df-dmqs 36731 | . . 3 ⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴) | |
3 | 2 | anbi2i 622 | . 2 ⊢ (( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴) ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) |
4 | 1, 3 | bitri 274 | 1 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1541 dom cdm 5588 / cqs 8471 EqvRel weqvrel 36329 DomainQs wdmqs 36336 ErALTV werALTV 36338 |
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 206 df-an 396 df-dmqs 36731 df-erALTV 36755 |
This theorem is referenced by: erALTVeq1 36760 dfmember2 36764 erim 36769 |
Copyright terms: Public domain | W3C validator |