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Theorem dfqs3 39351
 Description: Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
Assertion
Ref Expression
dfqs3 (𝐴 / 𝑅) = 𝑥𝐴 {[𝑥]𝑅}
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem dfqs3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-qs 8291 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
2 iunsn 39346 . 2 𝑥𝐴 {[𝑥]𝑅} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
31, 2eqtr4i 2850 1 (𝐴 / 𝑅) = 𝑥𝐴 {[𝑥]𝑅}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  {cab 2802  ∃wrex 3134  {csn 4550  ∪ ciun 4905  [cec 8283   / cqs 8284 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-rex 3139  df-v 3482  df-sn 4551  df-iun 4907  df-qs 8291 This theorem is referenced by:  prjspval2  39523
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