Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  qsresid Structured version   Visualization version   GIF version

Theorem qsresid 38307
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 38261 . . . . 5 (𝑣𝐴 → [𝑣](𝑅𝐴) = [𝑣]𝑅)
21eqeq2d 2746 . . . 4 (𝑣𝐴 → (𝑢 = [𝑣](𝑅𝐴) ↔ 𝑢 = [𝑣]𝑅))
32rexbiia 3090 . . 3 (∃𝑣𝐴 𝑢 = [𝑣](𝑅𝐴) ↔ ∃𝑣𝐴 𝑢 = [𝑣]𝑅)
43abbii 2807 . 2 {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣](𝑅𝐴)} = {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣]𝑅}
5 df-qs 8750 . 2 (𝐴 / (𝑅𝐴)) = {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣](𝑅𝐴)}
6 df-qs 8750 . 2 (𝐴 / 𝑅) = {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣]𝑅}
74, 5, 63eqtr4i 2773 1 (𝐴 / (𝑅𝐴)) = (𝐴 / 𝑅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  cres 5691  [cec 8742   / cqs 8743
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-10 2139  ax-11 2155  ax-12 2175  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-nf 1781  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-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-qs 8750
This theorem is referenced by:  n0elim  38632
  Copyright terms: Public domain W3C validator