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Theorem dfqs2 39438
 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 8281 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
2 eqid 2798 . . 3 (𝑥𝐴 ↦ [𝑥]𝑅) = (𝑥𝐴 ↦ [𝑥]𝑅)
32rnmpt 5792 . 2 ran (𝑥𝐴 ↦ [𝑥]𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
41, 3eqtr4i 2824 1 (𝐴 / 𝑅) = ran (𝑥𝐴 ↦ [𝑥]𝑅)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  {cab 2776  ∃wrex 3107   ↦ cmpt 5111  ran crn 5521  [cec 8273   / cqs 8274 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 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-mpt 5112  df-cnv 5528  df-dm 5530  df-rn 5531  df-qs 8281 This theorem is referenced by:  qsalrel  39441
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