Definition df-qs 7981

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 7974	. 2 class (𝐴 / 𝑅)
4		vy	. . . . . 6 setvar 𝑦
5	4	cv 1636	. . . . 5 class 𝑦
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1636	. . . . . 6 class 𝑥
8	7, 2	cec 7973	. . . . 5 class [𝑥]𝑅
9	5, 8	wceq 1637	. . . 4 wff 𝑦 = [𝑥]𝑅
10	9, 6, 1	wrex 3097	. . 3 wff ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅
11	10, 4	cab 2792	. 2 class {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
12	3, 11	wceq 1637	1 wff (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}

This definition is referenced by:  qseq1  8027  qseq2  8028  elqsg  8029  qsexg  8036  uniqs  8038  snec  8041  qsinxp  8054  qliftf  8066  quslem  16404  pi1xfrf  23062  pi1cof  23068  qsss1  34368  qsresid  34409  uniqsALTV  34414  0qs  34443