Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-ec Structured version   Visualization version   GIF version

Definition df-ec 7631
 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 7630). 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 7632. (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 7627 . 2 class [𝐴]𝑅
41csn 4125 . . 3 class {𝐴}
52, 4cima 5041 . 2 class (𝑅 “ {𝐴})
63, 5wceq 1475 1 wff [𝐴]𝑅 = (𝑅 “ {𝐴})
 Colors of variables: wff setvar class This definition is referenced by:  dfec2  7632  ecexg  7633  ecexr  7634  eceq1  7669  eceq2  7671  elecg  7672  ecss  7675  ecidsn  7682  uniqs  7694  ecqs  7698  ecinxp  7709
 Copyright terms: Public domain W3C validator