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Theorem epini 5043
Description: Any set is equal to its preimage under the converse epsilon relation. (Contributed by Mario Carneiro, 9-Mar-2013.)
Hypothesis
Ref Expression
epini.1  |-  A  e. 
_V
Assertion
Ref Expression
epini  |-  ( `'  _E  " { A } )  =  A

Proof of Theorem epini
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 epini.1 . . . 4  |-  A  e. 
_V
2 vex 2791 . . . . 5  |-  x  e. 
_V
32eliniseg 5042 . . . 4  |-  ( A  e.  _V  ->  (
x  e.  ( `'  _E  " { A } )  <->  x  _E  A ) )
41, 3ax-mp 8 . . 3  |-  ( x  e.  ( `'  _E  " { A } )  <-> 
x  _E  A )
51epelc 4307 . . 3  |-  ( x  _E  A  <->  x  e.  A )
64, 5bitri 240 . 2  |-  ( x  e.  ( `'  _E  " { A } )  <-> 
x  e.  A )
76eqriv 2280 1  |-  ( `'  _E  " { A } )  =  A
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   class class class wbr 4023    _E cep 4303   `'ccnv 4688   "cima 4692
This theorem is referenced by:  infxpenlem  7641  fz1isolem  11399
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-eprel 4305  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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