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Theorem erALTVeq1 38651
Description: Equality theorem for equivalence relation on domain quotient. (Contributed by Peter Mazsa, 25-Sep-2021.)
Assertion
Ref Expression
erALTVeq1 (𝑅 = 𝑆 → (𝑅 ErALTV 𝐴𝑆 ErALTV 𝐴))

Proof of Theorem erALTVeq1
StepHypRef 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))
31, 2anbi12d 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))
63, 4, 53bitr4g 314 1 (𝑅 = 𝑆 → (𝑅 ErALTV 𝐴𝑆 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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