ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  snec Unicode version

Theorem snec 6198
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 6172 . . . . 5  |-  ( x  =  A  ->  [ x ] R  =  [ A ] R )
32eqeq2d 2067 . . . 4  |-  ( x  =  A  ->  (
y  =  [ x ] R  <->  y  =  [ A ] R ) )
41, 3rexsn 3443 . . 3  |-  ( E. x  e.  { A } y  =  [
x ] R  <->  y  =  [ A ] R )
54abbii 2169 . 2  |-  { y  |  E. x  e. 
{ A } y  =  [ x ] R }  =  {
y  |  y  =  [ A ] R }
6 df-qs 6143 . 2  |-  ( { A } /. R
)  =  { y  |  E. x  e. 
{ A } y  =  [ x ] R }
7 df-sn 3409 . 2  |-  { [ A ] R }  =  { y  |  y  =  [ A ] R }
85, 6, 73eqtr4ri 2087 1  |-  { [ A ] R }  =  ( { A } /. R )
Colors of variables: wff set class
Syntax hints:    = wceq 1259    e. wcel 1409   {cab 2042   E.wrex 2324   _Vcvv 2574   {csn 3403   [cec 6135   /.cqs 6136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-ec 6139  df-qs 6143
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator