Theorem dfqs2 42235

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 8723	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
2		eqid 2735	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅) = (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅)
3	2	rnmpt 5937	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
4	1, 3	eqtr4i 2761	1 ⊢ (𝐴 / 𝑅) = ran (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅)

Syntax hints:   = wceq 1540  {cab 2713  ∃wrex 3060   ↦ cmpt 5201  ran crn 5655  [cec 8715   / cqs 8716

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-mpt 5202  df-cnv 5662  df-dm 5664  df-rn 5665  df-qs 8723

This theorem is referenced by:  qsalrel  42238