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Theorem epini 4746
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 2613 . . . . 5  |-  x  e. 
_V
32eliniseg 4745 . . . 4  |-  ( A  e.  _V  ->  (
x  e.  ( `'  _E  " { A } )  <->  x  _E  A ) )
41, 3ax-mp 7 . . 3  |-  ( x  e.  ( `'  _E  " { A } )  <-> 
x  _E  A )
51epelc 4074 . . 3  |-  ( x  _E  A  <->  x  e.  A )
64, 5bitri 182 . 2  |-  ( x  e.  ( `'  _E  " { A } )  <-> 
x  e.  A )
76eqriv 2080 1  |-  ( `'  _E  " { A } )  =  A
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1285    e. wcel 1434   _Vcvv 2610   {csn 3416   class class class wbr 3805    _E cep 4070   `'ccnv 4390   "cima 4394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-eprel 4072  df-xp 4397  df-cnv 4399  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404
This theorem is referenced by: (None)
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