Theorem snec 6830

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 6802	. . . . 5 |- ( x = A -> [ x ] R = [ A ] R )
3	2	eqeq2d 2244	. . . 4 |- ( x = A -> ( y = [ x ] R <-> y = [ A ] R ) )
4	1, 3	rexsn 3733	. . 3 |- ( E. x e. { A } y = [ x ] R <-> y = [ A ] R )
5	4	abbii 2348	. 2 |- { y | E. x e. { A } y = [ x ] R } = { y | y = [ A ] R }
6		df-qs 6773	. 2 |- ( { A } /. R ) = { y | E. x e. { A } y = [ x ] R }
7		df-sn 3695	. 2 |- { [ A ] R } = { y | y = [ A ] R }
8	5, 6, 7	3eqtr4ri 2264	1 |- { [ A ] R } = ( { A } /. R )

Syntax hints:   = wceq 1398   e. wcel 2203   {cab 2218   E.wrex 2521   _Vcvv 2813   {csn 3689   [cec 6765   /.cqs 6766

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-ec 6769  df-qs 6773

This theorem is referenced by: (None)