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Theorem snec 6926
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 6900 . . . . 5  |-  ( x  =  A  ->  [ x ] R  =  [ A ] R )
32eqeq2d 2415 . . . 4  |-  ( x  =  A  ->  (
y  =  [ x ] R  <->  y  =  [ A ] R ) )
41, 3rexsn 3810 . . 3  |-  ( E. x  e.  { A } y  =  [
x ] R  <->  y  =  [ A ] R )
54abbii 2516 . 2  |-  { y  |  E. x  e. 
{ A } y  =  [ x ] R }  =  {
y  |  y  =  [ A ] R }
6 df-qs 6870 . 2  |-  ( { A } /. R
)  =  { y  |  E. x  e. 
{ A } y  =  [ x ] R }
7 df-sn 3780 . 2  |-  { [ A ] R }  =  { y  |  y  =  [ A ] R }
85, 6, 73eqtr4ri 2435 1  |-  { [ A ] R }  =  ( { A } /. R )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   {cab 2390   E.wrex 2667   _Vcvv 2916   {csn 3774   [cec 6862   /.cqs 6863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ec 6866  df-qs 6870
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