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

Theorem qsid 6494
Description: A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
qsid  |-  ( A /. `'  _E  )  =  A

Proof of Theorem qsid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2689 . . . . . . 7  |-  x  e. 
_V
21ecid 6492 . . . . . 6  |-  [ x ] `'  _E  =  x
32eqeq2i 2150 . . . . 5  |-  ( y  =  [ x ] `'  _E  <->  y  =  x )
4 equcom 1682 . . . . 5  |-  ( y  =  x  <->  x  =  y )
53, 4bitri 183 . . . 4  |-  ( y  =  [ x ] `'  _E  <->  x  =  y
)
65rexbii 2442 . . 3  |-  ( E. x  e.  A  y  =  [ x ] `'  _E  <->  E. x  e.  A  x  =  y )
7 vex 2689 . . . 4  |-  y  e. 
_V
87elqs 6480 . . 3  |-  ( y  e.  ( A /. `'  _E  )  <->  E. x  e.  A  y  =  [ x ] `'  _E  )
9 risset 2463 . . 3  |-  ( y  e.  A  <->  E. x  e.  A  x  =  y )
106, 8, 93bitr4i 211 . 2  |-  ( y  e.  ( A /. `'  _E  )  <->  y  e.  A )
1110eqriv 2136 1  |-  ( A /. `'  _E  )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480   E.wrex 2417    _E cep 4209   `'ccnv 4538   [cec 6427   /.cqs 6428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-eprel 4211  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-ec 6431  df-qs 6435
This theorem is referenced by:  dfcnqs  7656
  Copyright terms: Public domain W3C validator