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Theorem epini 5236
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 2961 . . . . 5  |-  x  e. 
_V
32eliniseg 5235 . . . 4  |-  ( A  e.  _V  ->  (
x  e.  ( `'  _E  " { A } )  <->  x  _E  A ) )
41, 3ax-mp 8 . . 3  |-  ( x  e.  ( `'  _E  " { A } )  <-> 
x  _E  A )
51epelc 4498 . . 3  |-  ( x  _E  A  <->  x  e.  A )
64, 5bitri 242 . 2  |-  ( x  e.  ( `'  _E  " { A } )  <-> 
x  e.  A )
76eqriv 2435 1  |-  ( `'  _E  " { A } )  =  A
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816   class class class wbr 4214    _E cep 4494   `'ccnv 4879   "cima 4883
This theorem is referenced by:  infxpenlem  7897  fz1isolem  11712
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pr 4405
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 4215  df-opab 4269  df-eprel 4496  df-xp 4886  df-cnv 4888  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893
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