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Theorem erALTVeq1 38042
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 37975 . . 3 (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
2 dmqseqeq1 38016 . . 3 (𝑅 = 𝑆 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom 𝑆 / 𝑆) = 𝐴))
31, 2anbi12d 630 . 2 (𝑅 = 𝑆 → (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) ↔ ( EqvRel 𝑆 ∧ (dom 𝑆 / 𝑆) = 𝐴)))
4 dferALTV2 38041 . 2 (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
5 dferALTV2 38041 . 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 205  wa 395   = wceq 1533  dom cdm 5667   / cqs 8699   EqvRel weqvrel 37563   ErALTV werALTV 37572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ec 8702  df-qs 8706  df-refrel 37885  df-symrel 37917  df-trrel 37947  df-eqvrel 37958  df-dmqs 38012  df-erALTV 38037
This theorem is referenced by:  erALTVeq1i  38043  erALTVeq1d  38044
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