Theorem dfqs2 42256

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 8749	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
2		eqid 2734	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅) = (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅)
3	2	rnmpt 5970	. 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
4	1, 3	eqtr4i 2765	1 ⊢ (𝐴 / 𝑅) = ran (𝑥 ∈ 𝐴 ↦ [𝑥]𝑅)

Syntax hints:   = wceq 1536  {cab 2711  ∃wrex 3067   ↦ cmpt 5230  ran crn 5689  [cec 8741   / cqs 8742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-mpt 5231  df-cnv 5696  df-dm 5698  df-rn 5699  df-qs 8749

This theorem is referenced by:  qsalrel  42259