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Theorem qsid 6230
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 2605 . . . . . . 7  |-  x  e. 
_V
21ecid 6228 . . . . . 6  |-  [ x ] `'  _E  =  x
32eqeq2i 2092 . . . . 5  |-  ( y  =  [ x ] `'  _E  <->  y  =  x )
4 equcom 1634 . . . . 5  |-  ( y  =  x  <->  x  =  y )
53, 4bitri 182 . . . 4  |-  ( y  =  [ x ] `'  _E  <->  x  =  y
)
65rexbii 2374 . . 3  |-  ( E. x  e.  A  y  =  [ x ] `'  _E  <->  E. x  e.  A  x  =  y )
7 vex 2605 . . . 4  |-  y  e. 
_V
87elqs 6216 . . 3  |-  ( y  e.  ( A /. `'  _E  )  <->  E. x  e.  A  y  =  [ x ] `'  _E  )
9 risset 2395 . . 3  |-  ( y  e.  A  <->  E. x  e.  A  x  =  y )
106, 8, 93bitr4i 210 . 2  |-  ( y  e.  ( A /. `'  _E  )  <->  y  e.  A )
1110eqriv 2079 1  |-  ( A /. `'  _E  )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434   E.wrex 2350    _E cep 4044   `'ccnv 4364   [cec 6163   /.cqs 6164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-eprel 4046  df-xp 4371  df-cnv 4373  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-ec 6167  df-qs 6171
This theorem is referenced by:  dfcnqs  7060
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