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Theorem qsresid 35463
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 35417 . . . . 5 (𝑣𝐴 → [𝑣](𝑅𝐴) = [𝑣]𝑅)
21eqeq2d 2829 . . . 4 (𝑣𝐴 → (𝑢 = [𝑣](𝑅𝐴) ↔ 𝑢 = [𝑣]𝑅))
32rexbiia 3243 . . 3 (∃𝑣𝐴 𝑢 = [𝑣](𝑅𝐴) ↔ ∃𝑣𝐴 𝑢 = [𝑣]𝑅)
43abbii 2883 . 2 {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣](𝑅𝐴)} = {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣]𝑅}
5 df-qs 8284 . 2 (𝐴 / (𝑅𝐴)) = {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣](𝑅𝐴)}
6 df-qs 8284 . 2 (𝐴 / 𝑅) = {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣]𝑅}
74, 5, 63eqtr4i 2851 1 (𝐴 / (𝑅𝐴)) = (𝐴 / 𝑅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  {cab 2796  wrex 3136  cres 5550  [cec 8276   / cqs 8277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ec 8280  df-qs 8284
This theorem is referenced by:  n0el3  35765
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