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Mirrors > Home > MPE Home > Th. List > Mathboxes > erALTVeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for equivalence relation on domain quotient. (Contributed by Peter Mazsa, 25-Sep-2021.) |
Ref | Expression |
---|---|
erALTVeq1 | ⊢ (𝑅 = 𝑆 → (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvreleq 38584 | . . 3 ⊢ (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆)) | |
2 | dmqseqeq1 38625 | . . 3 ⊢ (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴)) | |
3 | 1, 2 | anbi12d 632 | . 2 ⊢ (𝑅 = 𝑆 → (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( EqvRel 𝑆 ∧ (dom 𝑆 / 𝑆) = 𝐴))) |
4 | dferALTV2 38650 | . 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) | |
5 | dferALTV2 38650 | . 2 ⊢ (𝑆 ErALTV 𝐴 ↔ ( EqvRel 𝑆 ∧ (dom 𝑆 / 𝑆) = 𝐴)) | |
6 | 3, 4, 5 | 3bitr4g 314 | 1 ⊢ (𝑅 = 𝑆 → (𝑅 ErALTV 𝐴 ↔ 𝑆 ErALTV 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 dom cdm 5689 / cqs 8743 EqvRel weqvrel 38179 ErALTV werALTV 38188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ec 8746 df-qs 8750 df-refrel 38494 df-symrel 38526 df-trrel 38556 df-eqvrel 38567 df-dmqs 38621 df-erALTV 38646 |
This theorem is referenced by: erALTVeq1i 38652 erALTVeq1d 38653 |
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