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Metamath Proof Explorer

Definition df-ec

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 ). 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 . (Contributed by NM, 23-Jul-1995)

		Ref	Expression
	Assertion	df-ec	$⊢ [A] R = R [\{A\}]$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cA	$class A$
1		cR	$class R$
2	0 1	cec	$class [A] R$
3	0	csn	$class \{A\}$
4	1 3	cima	$class R [\{A\}]$
5	2 4	wceq	$wff [A] R = R [\{A\}]$