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Theorem snec 6722
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 6696 . . . . 5  |-  ( x  =  A  ->  [ x ] R  =  [ A ] R )
32eqeq2d 2294 . . . 4  |-  ( x  =  A  ->  (
y  =  [ x ] R  <->  y  =  [ A ] R ) )
41, 3rexsn 3675 . . 3  |-  ( E. x  e.  { A } y  =  [
x ] R  <->  y  =  [ A ] R )
54abbii 2395 . 2  |-  { y  |  E. x  e. 
{ A } y  =  [ x ] R }  =  {
y  |  y  =  [ A ] R }
6 df-qs 6666 . 2  |-  ( { A } /. R
)  =  { y  |  E. x  e. 
{ A } y  =  [ x ] R }
7 df-sn 3646 . 2  |-  { [ A ] R }  =  { y  |  y  =  [ A ] R }
85, 6, 73eqtr4ri 2314 1  |-  { [ A ] R }  =  ( { A } /. R )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   {csn 3640   [cec 6658   /.cqs 6659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-ec 6662  df-qs 6666
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