Theorem dfqs3 43383

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 8726	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
2		iunsn 5023	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
3	1, 2	eqtr4i 2786	1 ⊢ (𝐴 / 𝑅) = ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅}

This proof depends on syntax axioms:   = wceq 1569  {cab 2738  ∃wrex 3086  {csn 4584  ∪ ciun 4950  [cec 8718   / cqs 8719

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rex 3087  df-v 3452  df-sn 4585  df-iun 4952  df-qs 8726

This theorem is used by:  prjspval2  43734