Definition df-ec 6813

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 6812). 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 6814. (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 6809	. 2 class [𝐴]𝑅
4	1	csn 3709	. . 3 class {𝐴}
5	2, 4	cima 4779	. 2 class (𝑅 “ {𝐴})
6	3, 5	wceq 1402	1 wff [𝐴]𝑅 = (𝑅 “ {𝐴})

This definition is used by:  dfec2  6814  ecexg  6815  ecexr  6816  eceq1  6846  eceq2  6848  elecg  6851  ecss  6854  ecidsn  6860  uniqs  6871  ecqs  6875  ecinxp  6888