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Theorem dfqs2 41365
Description: Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
Assertion
Ref Expression
dfqs2 (𝐴 / 𝑅) = ran (𝑥𝐴 ↦ [𝑥]𝑅)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem dfqs2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-qs 8711 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
2 eqid 2732 . . 3 (𝑥𝐴 ↦ [𝑥]𝑅) = (𝑥𝐴 ↦ [𝑥]𝑅)
32rnmpt 5954 . 2 ran (𝑥𝐴 ↦ [𝑥]𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
41, 3eqtr4i 2763 1 (𝐴 / 𝑅) = ran (𝑥𝐴 ↦ [𝑥]𝑅)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  {cab 2709  wrex 3070  cmpt 5231  ran crn 5677  [cec 8703   / cqs 8704
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rex 3071  df-rab 3433  df-v 3476  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-mpt 5232  df-cnv 5684  df-dm 5686  df-rn 5687  df-qs 8711
This theorem is referenced by:  qsalrel  41368
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