Theorem dfqs3 42233

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 8769	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
2		iunsn 5089	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
3	1, 2	eqtr4i 2771	1 ⊢ (𝐴 / 𝑅) = ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅}

Syntax hints:   = wceq 1537  {cab 2717  ∃wrex 3076  {csn 4648  ∪ ciun 5015  [cec 8761   / cqs 8762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rex 3077  df-v 3490  df-sn 4649  df-iun 5017  df-qs 8769

This theorem is referenced by:  prjspval2  42568