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Theorem qsid 7334
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 3061 . . . . . . 7  |-  x  e. 
_V
21ecid 7333 . . . . . 6  |-  [ x ] `'  _E  =  x
32eqeq2i 2420 . . . . 5  |-  ( y  =  [ x ] `'  _E  <->  y  =  x )
4 equcom 1818 . . . . 5  |-  ( y  =  x  <->  x  =  y )
53, 4bitri 249 . . . 4  |-  ( y  =  [ x ] `'  _E  <->  x  =  y
)
65rexbii 2905 . . 3  |-  ( E. x  e.  A  y  =  [ x ] `'  _E  <->  E. x  e.  A  x  =  y )
7 vex 3061 . . . 4  |-  y  e. 
_V
87elqs 7321 . . 3  |-  ( y  e.  ( A /. `'  _E  )  <->  E. x  e.  A  y  =  [ x ] `'  _E  )
9 risset 2931 . . 3  |-  ( y  e.  A  <->  E. x  e.  A  x  =  y )
106, 8, 93bitr4i 277 . 2  |-  ( y  e.  ( A /. `'  _E  )  <->  y  e.  A )
1110eqriv 2398 1  |-  ( A /. `'  _E  )  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    e. wcel 1842   E.wrex 2754    _E cep 4731   `'ccnv 4941   [cec 7266   /.cqs 7267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-eprel 4733  df-xp 4948  df-cnv 4950  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-ec 7270  df-qs 7274
This theorem is referenced by:  dfcnqs  9469
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