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Theorem qsid 6973
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 2961 . . . . . . 7  |-  x  e. 
_V
21ecid 6972 . . . . . 6  |-  [ x ] `'  _E  =  x
32eqeq2i 2448 . . . . 5  |-  ( y  =  [ x ] `'  _E  <->  y  =  x )
4 equcom 1693 . . . . 5  |-  ( y  =  x  <->  x  =  y )
53, 4bitri 242 . . . 4  |-  ( y  =  [ x ] `'  _E  <->  x  =  y
)
65rexbii 2732 . . 3  |-  ( E. x  e.  A  y  =  [ x ] `'  _E  <->  E. x  e.  A  x  =  y )
7 vex 2961 . . . 4  |-  y  e. 
_V
87elqs 6960 . . 3  |-  ( y  e.  ( A /. `'  _E  )  <->  E. x  e.  A  y  =  [ x ] `'  _E  )
9 risset 2755 . . 3  |-  ( y  e.  A  <->  E. x  e.  A  x  =  y )
106, 8, 93bitr4i 270 . 2  |-  ( y  e.  ( A /. `'  _E  )  <->  y  e.  A )
1110eqriv 2435 1  |-  ( A /. `'  _E  )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   E.wrex 2708    _E cep 4495   `'ccnv 4880   [cec 6906   /.cqs 6907
This theorem is referenced by:  dfcnqs  9022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-eprel 4497  df-xp 4887  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-ec 6910  df-qs 6914
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