Theorem snecg

Description: The singleton of a coset is the singleton quotient. (Contributed by Peter Mazsa, 25-Mar-2019)

	Ref	Expression
Assertion	snecg	|- ( A e. V -> { [ A ] R } = ( { A } /. R ) )

Proof

Step	Hyp	Ref	Expression
1		eceq1	|- ( x = A -> [ x ] R = [ A ] R )
2	1	eqeq2d	|- ( x = A -> ( y = [ x ] R <-> y = [ A ] R ) )
3	2	rexsng	|- ( A e. V -> ( E. x e. { A } y = [ x ] R <-> y = [ A ] R ) )
4	3	abbidv	|- ( A e. V -> { y | E. x e. { A } y = [ x ] R } = { y | y = [ A ] R } )
5		df-qs	|- ( { A } /. R ) = { y | E. x e. { A } y = [ x ] R }
6		df-sn	|- { [ A ] R } = { y | y = [ A ] R }
7	4 5 6	3eqtr4g	|- ( A e. V -> ( { A } /. R ) = { [ A ] R } )
8	7	eqcomd	|- ( A e. V -> { [ A ] R } = ( { A } /. R ) )