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Theorem erALTVeq1 37181
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 37114 . . 3 (𝑅 = 𝑆 → ( EqvRel 𝑅 ↔ EqvRel 𝑆))
2 dmqseqeq1 37155 . . 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 37180 . 2 (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
5 dferALTV2 37180 . 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 397   = wceq 1542  dom cdm 5637   / cqs 8653   EqvRel weqvrel 36701   ErALTV werALTV 36710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ec 8656  df-qs 8660  df-refrel 37024  df-symrel 37056  df-trrel 37086  df-eqvrel 37097  df-dmqs 37151  df-erALTV 37176
This theorem is referenced by:  erALTVeq1i  37182  erALTVeq1d  37183
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