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Theorem qsresid 36048
 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 36002 . . . . 5 (𝑣𝐴 → [𝑣](𝑅𝐴) = [𝑣]𝑅)
21eqeq2d 2769 . . . 4 (𝑣𝐴 → (𝑢 = [𝑣](𝑅𝐴) ↔ 𝑢 = [𝑣]𝑅))
32rexbiia 3174 . . 3 (∃𝑣𝐴 𝑢 = [𝑣](𝑅𝐴) ↔ ∃𝑣𝐴 𝑢 = [𝑣]𝑅)
43abbii 2823 . 2 {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣](𝑅𝐴)} = {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣]𝑅}
5 df-qs 8310 . 2 (𝐴 / (𝑅𝐴)) = {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣](𝑅𝐴)}
6 df-qs 8310 . 2 (𝐴 / 𝑅) = {𝑢 ∣ ∃𝑣𝐴 𝑢 = [𝑣]𝑅}
74, 5, 63eqtr4i 2791 1 (𝐴 / (𝑅𝐴)) = (𝐴 / 𝑅)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  {cab 2735  ∃wrex 3071   ↾ cres 5529  [cec 8302   / cqs 8303 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5036  df-opab 5098  df-xp 5533  df-rel 5534  df-cnv 5535  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-ec 8306  df-qs 8310 This theorem is referenced by:  n0el3  36351
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