Theorem dfqs3 43048

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 8709	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
2		iunsn 5035	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
3	1, 2	eqtr4i 2792	1 ⊢ (𝐴 / 𝑅) = ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅}

This proof depends on syntax axioms:   = wceq 1570  {cab 2744  ∃wrex 3092  {csn 4594  ∪ ciun 4961  [cec 8701   / cqs 8702

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rex 3093  df-v 3460  df-sn 4595  df-iun 4963  df-qs 8709

This theorem is used by:  prjspval2  43386