Theorem snec 6596

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 6570	. . . . 5 |- ( x = A -> [ x ] R = [ A ] R )
3	2	eqeq2d 2189	. . . 4 |- ( x = A -> ( y = [ x ] R <-> y = [ A ] R ) )
4	1, 3	rexsn 3637	. . 3 |- ( E. x e. { A } y = [ x ] R <-> y = [ A ] R )
5	4	abbii 2293	. 2 |- { y | E. x e. { A } y = [ x ] R } = { y | y = [ A ] R }
6		df-qs 6541	. 2 |- ( { A } /. R ) = { y | E. x e. { A } y = [ x ] R }
7		df-sn 3599	. 2 |- { [ A ] R } = { y | y = [ A ] R }
8	5, 6, 7	3eqtr4ri 2209	1 |- { [ A ] R } = ( { A } /. R )

Syntax hints:   = wceq 1353   e. wcel 2148   {cab 2163   E.wrex 2456   _Vcvv 2738   {csn 3593   [cec 6533   /.cqs 6534

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-xp 4633  df-cnv 4635  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-ec 6537  df-qs 6541

This theorem is referenced by: (None)