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Metamath Proof Explorer

Theorem dfec2

Description: Alternate definition of R -coset of A . Definition 34 of Suppes p. 81. (Contributed by NM, 3-Jan-1997) (Proof shortened by Mario Carneiro, 9-Jul-2014)

		Ref	Expression
	Assertion	dfec2	⊢ ( 𝐴 ∈ 𝑉 → [ 𝐴 ] 𝑅 = { 𝑦 ∣ 𝐴 𝑅 𝑦 } )

Proof

Step	Hyp	Ref	Expression
1		df-ec	⊢ [ 𝐴 ] 𝑅 = ( 𝑅 “ { 𝐴 } )
2		imasng	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑅 “ { 𝐴 } ) = { 𝑦 ∣ 𝐴 𝑅 𝑦 } )
3	1 2	eqtrid	⊢ ( 𝐴 ∈ 𝑉 → [ 𝐴 ] 𝑅 = { 𝑦 ∣ 𝐴 𝑅 𝑦 } )