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Theorem qsid 6462
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 2663 . . . . . . 7  |-  x  e. 
_V
21ecid 6460 . . . . . 6  |-  [ x ] `'  _E  =  x
32eqeq2i 2128 . . . . 5  |-  ( y  =  [ x ] `'  _E  <->  y  =  x )
4 equcom 1667 . . . . 5  |-  ( y  =  x  <->  x  =  y )
53, 4bitri 183 . . . 4  |-  ( y  =  [ x ] `'  _E  <->  x  =  y
)
65rexbii 2419 . . 3  |-  ( E. x  e.  A  y  =  [ x ] `'  _E  <->  E. x  e.  A  x  =  y )
7 vex 2663 . . . 4  |-  y  e. 
_V
87elqs 6448 . . 3  |-  ( y  e.  ( A /. `'  _E  )  <->  E. x  e.  A  y  =  [ x ] `'  _E  )
9 risset 2440 . . 3  |-  ( y  e.  A  <->  E. x  e.  A  x  =  y )
106, 8, 93bitr4i 211 . 2  |-  ( y  e.  ( A /. `'  _E  )  <->  y  e.  A )
1110eqriv 2114 1  |-  ( A /. `'  _E  )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465   E.wrex 2394    _E cep 4179   `'ccnv 4508   [cec 6395   /.cqs 6396
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-eprel 4181  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-ec 6399  df-qs 6403
This theorem is referenced by:  dfcnqs  7617
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