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Metamath Proof Explorer

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 ) )