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Theorem snec 6738
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.
StepHypRef Expression
1 snec.1 . . . 4  |-  A  e. 
_V
2 eceq1 6712 . . . . 5  |-  ( x  =  A  ->  [ x ] R  =  [ A ] R )
32eqeq2d 2307 . . . 4  |-  ( x  =  A  ->  (
y  =  [ x ] R  <->  y  =  [ A ] R ) )
41, 3rexsn 3688 . . 3  |-  ( E. x  e.  { A } y  =  [
x ] R  <->  y  =  [ A ] R )
54abbii 2408 . 2  |-  { y  |  E. x  e. 
{ A } y  =  [ x ] R }  =  {
y  |  y  =  [ A ] R }
6 df-qs 6682 . 2  |-  ( { A } /. R
)  =  { y  |  E. x  e. 
{ A } y  =  [ x ] R }
7 df-sn 3659 . 2  |-  { [ A ] R }  =  { y  |  y  =  [ A ] R }
85, 6, 73eqtr4ri 2327 1  |-  { [ A ] R }  =  ( { A } /. R )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801   {csn 3653   [cec 6674   /.cqs 6675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-ec 6678  df-qs 6682
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