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Theorem snec 6967
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 6941 . . . . 5  |-  ( x  =  A  ->  [ x ] R  =  [ A ] R )
32eqeq2d 2447 . . . 4  |-  ( x  =  A  ->  (
y  =  [ x ] R  <->  y  =  [ A ] R ) )
41, 3rexsn 3850 . . 3  |-  ( E. x  e.  { A } y  =  [
x ] R  <->  y  =  [ A ] R )
54abbii 2548 . 2  |-  { y  |  E. x  e. 
{ A } y  =  [ x ] R }  =  {
y  |  y  =  [ A ] R }
6 df-qs 6911 . 2  |-  ( { A } /. R
)  =  { y  |  E. x  e. 
{ A } y  =  [ x ] R }
7 df-sn 3820 . 2  |-  { [ A ] R }  =  { y  |  y  =  [ A ] R }
85, 6, 73eqtr4ri 2467 1  |-  { [ A ] R }  =  ( { A } /. R )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   {cab 2422   E.wrex 2706   _Vcvv 2956   {csn 3814   [cec 6903   /.cqs 6904
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-ec 6907  df-qs 6911
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