Theorem dfqs2 42278

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

Assertion

Ref	Expression
dfqs2	⊢ (𝐴 / 𝑅) = ran (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem dfqs2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-qs 8751	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
2		eqid 2737	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅) = (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅)
3	2	rnmpt 5968	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
4	1, 3	eqtr4i 2768	1 ⊢ (𝐴 / 𝑅) = ran (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅)

Syntax hints:   = wceq 1540  {cab 2714  ∃wrex 3070   ↦ cmpt 5225  ran crn 5686  [cec 8743   / cqs 8744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-cnv 5693  df-dm 5695  df-rn 5696  df-qs 8751

This theorem is referenced by:  qsalrel  42281