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Theorem dfqs3 42257
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 8749 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
2 iunsn 5070 . 2 𝑥𝐴 {[𝑥]𝑅} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
31, 2eqtr4i 2765 1 (𝐴 / 𝑅) = 𝑥𝐴 {[𝑥]𝑅}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  {cab 2711  wrex 3067  {csn 4630   ciun 4995  [cec 8741   / cqs 8742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rex 3068  df-v 3479  df-sn 4631  df-iun 4997  df-qs 8749
This theorem is referenced by:  prjspval2  42599
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