Definition df-ec 6044

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 6043). 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 6045. (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 6040	. 2 class [A]𝑅
4	1	csn 3367	. . 3 class {A}
5	2, 4	cima 4291	. 2 class (𝑅 “ {A})
6	3, 5	wceq 1242	1 wff [A]𝑅 = (𝑅 “ {A})

This definition is referenced by:  dfec2  6045  ecexg  6046  ecexr  6047  eceq1  6077  eceq2  6079  elecg  6080  ecss  6083  ecidsn  6089  uniqs  6100  ecqs  6104  ecinxp  6117