Theorem dfqs3 40213

Description: Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)

Assertion

Ref	Expression
dfqs3	⊢ (𝐴 / 𝑅) = ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅}

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem dfqs3

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-qs 8504	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
2		iunsn 4995	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
3	1, 2	eqtr4i 2769	1 ⊢ (𝐴 / 𝑅) = ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅}

Syntax hints:   = wceq 1539  {cab 2715  ∃wrex 3065  {csn 4561  ∪ ciun 4924  [cec 8496   / cqs 8497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rex 3070  df-v 3434  df-sn 4562  df-iun 4926  df-qs 8504

This theorem is referenced by:  prjspval2  40452