Definition df-qs 7748

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 7741	. 2 class (𝐴 / 𝑅)
4		vy	. . . . . 6 setvar 𝑦
5	4	cv 1482	. . . . 5 class 𝑦
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1482	. . . . . 6 class 𝑥
8	7, 2	cec 7740	. . . . 5 class [𝑥]𝑅
9	5, 8	wceq 1483	. . . 4 wff 𝑦 = [𝑥]𝑅
10	9, 6, 1	wrex 2913	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅
11	10, 4	cab 2608	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
12	3, 11	wceq 1483	1 wff (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}

This definition is referenced by:  qseq1  7796  qseq2  7797  elqsg  7798  qsexg  7805  uniqs  7807  snec  7810  qsinxp  7823  qliftf  7835  quslem  16203  pi1xfrf  22853  pi1cof  22859  qsss1  34053  qsresid  34096  uniqsALTV  34101  0qs  34133