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Definition df-ec 7789
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 7788). 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 7790. (Contributed by NM, 23-Jul-1995.)
Assertion
Ref Expression
df-ec [𝐴]𝑅 = (𝑅 “ {𝐴})

Detailed syntax breakdown of Definition df-ec
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cR . . 3 class 𝑅
31, 2cec 7785 . 2 class [𝐴]𝑅
41csn 4210 . . 3 class {𝐴}
52, 4cima 5146 . 2 class (𝑅 “ {𝐴})
63, 5wceq 1523 1 wff [𝐴]𝑅 = (𝑅 “ {𝐴})
Colors of variables: wff setvar class
This definition is referenced by:  dfec2  7790  ecexg  7791  ecexr  7792  eceq1  7825  eceq2  7827  elecg  7828  ecss  7831  ecidsn  7838  uniqs  7850  ecqs  7854  ecinxp  7865  elecALTV  34171  uniqsALTV  34242  ec0  34271
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