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Theorem qsresid 37657
Description: Simplification of a special quotient set. (Contributed by Peter Mazsa, 2-Sep-2020.)
Assertion
Ref Expression
qsresid (𝐴 / (𝑅𝐴)) = (𝐴 / 𝑅)

Proof of Theorem qsresid
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecres2 37610 . . . . 5 (𝑣𝐴 → [𝑣](𝑅𝐴) = [𝑣]𝑅)
21eqeq2d 2742 . . . 4 (𝑣𝐴 → (𝑢 = [𝑣](𝑅𝐴) ↔ 𝑢 = [𝑣]𝑅))
32rexbiia 3091 . . 3 (∃𝑣𝐴 𝑢 = [𝑣](𝑅𝐴) ↔ ∃𝑣𝐴 𝑢 = [𝑣]𝑅)
43abbii 2801 . 2 {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣](𝑅𝐴)} = {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣]𝑅}
5 df-qs 8715 . 2 (𝐴 / (𝑅𝐴)) = {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣](𝑅𝐴)}
6 df-qs 8715 . 2 (𝐴 / 𝑅) = {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣]𝑅}
74, 5, 63eqtr4i 2769 1 (𝐴 / (𝑅𝐴)) = (𝐴 / 𝑅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2105  {cab 2708  wrex 3069  cres 5678  [cec 8707   / cqs 8708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8711  df-qs 8715
This theorem is referenced by:  n0elim  37983
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