Theorem snec 6743

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 6715	. . . . 5 |- ( x = A -> [ x ] R = [ A ] R )
3	2	eqeq2d 2241	. . . 4 |- ( x = A -> ( y = [ x ] R <-> y = [ A ] R ) )
4	1, 3	rexsn 3710	. . 3 |- ( E. x e. { A } y = [ x ] R <-> y = [ A ] R )
5	4	abbii 2345	. 2 |- { y | E. x e. { A } y = [ x ] R } = { y | y = [ A ] R }
6		df-qs 6686	. 2 |- ( { A } /. R ) = { y | E. x e. { A } y = [ x ] R }
7		df-sn 3672	. 2 |- { [ A ] R } = { y | y = [ A ] R }
8	5, 6, 7	3eqtr4ri 2261	1 |- { [ A ] R } = ( { A } /. R )

Syntax hints:   = wceq 1395   e. wcel 2200   {cab 2215   E.wrex 2509   _Vcvv 2799   {csn 3666   [cec 6678   /.cqs 6679

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-ec 6682  df-qs 6686

This theorem is referenced by: (None)