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Theorem dfqs3 41368
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 8713 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
2 iunsn 5070 . 2 𝑥𝐴 {[𝑥]𝑅} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
31, 2eqtr4i 2761 1 (𝐴 / 𝑅) = 𝑥𝐴 {[𝑥]𝑅}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  {cab 2707  wrex 3068  {csn 4629   ciun 4998  [cec 8705   / cqs 8706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rex 3069  df-v 3474  df-sn 4630  df-iun 5000  df-qs 8713
This theorem is referenced by:  prjspval2  41659
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