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Theorem coepr 24935
Description: Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
Hypotheses
Ref Expression
coep.1  |-  A  e. 
_V
coep.2  |-  B  e. 
_V
Assertion
Ref Expression
coepr  |-  ( A ( R  o.  `'  _E  ) B  <->  E. x  e.  A  x R B )
Distinct variable groups:    x, A    x, B    x, R

Proof of Theorem coepr
StepHypRef Expression
1 coep.1 . . . . . 6  |-  A  e. 
_V
2 vex 2876 . . . . . 6  |-  x  e. 
_V
31, 2brcnv 4967 . . . . 5  |-  ( A `'  _E  x  <->  x  _E  A )
41epelc 4410 . . . . 5  |-  ( x  _E  A  <->  x  e.  A )
53, 4bitri 240 . . . 4  |-  ( A `'  _E  x  <->  x  e.  A )
65anbi1i 676 . . 3  |-  ( ( A `'  _E  x  /\  x R B )  <-> 
( x  e.  A  /\  x R B ) )
76exbii 1587 . 2  |-  ( E. x ( A `'  _E  x  /\  x R B )  <->  E. x
( x  e.  A  /\  x R B ) )
8 coep.2 . . 3  |-  B  e. 
_V
91, 8brco 4955 . 2  |-  ( A ( R  o.  `'  _E  ) B  <->  E. x
( A `'  _E  x  /\  x R B ) )
10 df-rex 2634 . 2  |-  ( E. x  e.  A  x R B  <->  E. x
( x  e.  A  /\  x R B ) )
117, 9, 103bitr4i 268 1  |-  ( A ( R  o.  `'  _E  ) B  <->  E. x  e.  A  x R B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1546    e. wcel 1715   E.wrex 2629   _Vcvv 2873   class class class wbr 4125    _E cep 4406   `'ccnv 4791    o. ccom 4796
This theorem is referenced by:  elfuns  25280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-eprel 4408  df-cnv 4800  df-co 4801
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