Theorem snec 6652

Description: The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypothesis

Ref	Expression
snec.1	|- A e. _V

Assertion

Ref	Expression
snec	|- { [ A ] R } = ( { A } /. R )

Proof of Theorem snec

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snec.1	. . . 4 |- A e. _V
2		eceq1 6624	. . . . 5 |- ( x = A -> [ x ] R = [ A ] R )
3	2	eqeq2d 2205	. . . 4 |- ( x = A -> ( y = [ x ] R <-> y = [ A ] R ) )
4	1, 3	rexsn 3663	. . 3 |- ( E. x e. { A } y = [ x ] R <-> y = [ A ] R )
5	4	abbii 2309	. 2 |- { y | E. x e. { A } y = [ x ] R } = { y | y = [ A ] R }
6		df-qs 6595	. 2 |- ( { A } /. R ) = { y | E. x e. { A } y = [ x ] R }
7		df-sn 3625	. 2 |- { [ A ] R } = { y | y = [ A ] R }
8	5, 6, 7	3eqtr4ri 2225	1 |- { [ A ] R } = ( { A } /. R )

Syntax hints:   = wceq 1364   e. wcel 2164   {cab 2179   E.wrex 2473   _Vcvv 2760   {csn 3619   [cec 6587   /.cqs 6588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-cnv 4668  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-ec 6591  df-qs 6595

This theorem is referenced by: (None)