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Theorem dfqs3 40872
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 8692 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
2 iunsn 5062 . 2 𝑥𝐴 {[𝑥]𝑅} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
31, 2eqtr4i 2762 1 (𝐴 / 𝑅) = 𝑥𝐴 {[𝑥]𝑅}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  {cab 2708  wrex 3069  {csn 4622   ciun 4990  [cec 8684   / cqs 8685
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-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rex 3070  df-v 3475  df-sn 4623  df-iun 4992  df-qs 8692
This theorem is referenced by:  prjspval2  41137
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