Definition df-qs 7745

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 7738	. 2 class (𝐴 / 𝑅)
4		vy	. . . . . 6 setvar 𝑦
5	4	cv 1481	. . . . 5 class 𝑦
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1481	. . . . . 6 class 𝑥
8	7, 2	cec 7737	. . . . 5 class [𝑥]𝑅
9	5, 8	wceq 1482	. . . 4 wff 𝑦 = [𝑥]𝑅
10	9, 6, 1	wrex 2912	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅
11	10, 4	cab 2607	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
12	3, 11	wceq 1482	1 wff (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}

This definition is referenced by:  qseq1  7793  qseq2  7794  elqsg  7795  qsexg  7802  uniqs  7804  snec  7807  qsinxp  7820  qliftf  7832  quslem  16197  pi1xfrf  22847  pi1cof  22853