Definition df-ec 8673

Description: Define the 𝑅-coset of 𝐴. Exercise 35 of [Enderton] p. 61. This is called the equivalence class of 𝐴 modulo 𝑅 when 𝑅 is an equivalence relation (i.e. when Er 𝑅; see dfer2 8672). In this case, 𝐴 is a representative (member) of the equivalence class [𝐴]𝑅, which contains all sets that are equivalent to 𝐴. Definition of [Enderton] p. 57 uses the notation [𝐴] (subscript) 𝑅, although we simply follow the brackets by 𝑅 since we don't have subscripted expressions. For an alternate definition, see dfec2 8674. (Contributed by NM, 23-Jul-1995.)

Assertion

Ref	Expression
df-ec	⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})

Detailed syntax breakdown of Definition df-ec

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cR	. . 3 class 𝑅
3	1, 2	cec 8669	. 2 class [𝐴]𝑅
4	1	csn 4589	. . 3 class {𝐴}
5	2, 4	cima 5641	. 2 class (𝑅 “ {𝐴})
6	3, 5	wceq 1540	1 wff [𝐴]𝑅 = (𝑅 “ {𝐴})

This definition is referenced by:  dfec2  8674  ecexg  8675  ecexr  8676  eceq1  8710  eceq2  8712  elecg  8715  ecss  8722  ecidsn  8729  uniqs  8747  ecqs  8752  ecinxp  8765  eqg0subgecsn  19129  lsmsnorb  33362  elecALTV  38255  ec0  38351  prjspeclsp  42600