Theorem ecqs

Description: Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999)

	Ref	Expression
Hypothesis	ecqs.1	⊢ 𝑅 ∈ V
Assertion	ecqs	⊢ [ 𝐴 ] 𝑅 = ∪ ( { 𝐴 } / 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		ecqs.1	⊢ 𝑅 ∈ V
2		df-ec	⊢ [ 𝐴 ] 𝑅 = ( 𝑅 “ { 𝐴 } )
3		uniqsw	⊢ ( 𝑅 ∈ V → ∪ ( { 𝐴 } / 𝑅 ) = ( 𝑅 “ { 𝐴 } ) )
4	1 3	ax-mp	⊢ ∪ ( { 𝐴 } / 𝑅 ) = ( 𝑅 “ { 𝐴 } )
5	2 4	eqtr4i	⊢ [ 𝐴 ] 𝑅 = ∪ ( { 𝐴 } / 𝑅 )