Definition df-qs 8723

Description: Define quotient set. 𝑅 is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.)

Assertion

Ref	Expression
df-qs	⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦

Detailed syntax breakdown of Definition df-qs

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cR	. . 3 class 𝑅
3	1, 2	cqs 8716	. 2 class (𝐴 / 𝑅)
4		vy	. . . . . 6 setvar 𝑦
5	4	cv 1539	. . . . 5 class 𝑦
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1539	. . . . . 6 class 𝑥
8	7, 2	cec 8715	. . . . 5 class [𝑥]𝑅
9	5, 8	wceq 1540	. . . 4 wff 𝑦 = [𝑥]𝑅
10	9, 6, 1	wrex 3060	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅
11	10, 4	cab 2713	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
12	3, 11	wceq 1540	1 wff (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}

This definition is referenced by:  qseq1  8773  qseq2  8774  0qs  8779  elqsg  8780  qsexg  8787  uniqs  8789  snec  8792  qsinxp  8805  qliftf  8817  quslem  17555  qus0subgbas  19179  pzriprnglem11  21450  pi1xfrf  25002  pi1cof  25008  qusbas2  33367  qsss1  38253  qsresid  38289  uniqsALTV  38293  disjdmqscossss  38767  dfqs2  42235  dfqs3  42236