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Theorem coepr 25327
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 2923 . . . . . 6  |-  x  e. 
_V
31, 2brcnv 5018 . . . . 5  |-  ( A `'  _E  x  <->  x  _E  A )
41epelc 4460 . . . . 5  |-  ( x  _E  A  <->  x  e.  A )
53, 4bitri 241 . . . 4  |-  ( A `'  _E  x  <->  x  e.  A )
65anbi1i 677 . . 3  |-  ( ( A `'  _E  x  /\  x R B )  <-> 
( x  e.  A  /\  x R B ) )
76exbii 1589 . 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 5006 . 2  |-  ( A ( R  o.  `'  _E  ) B  <->  E. x
( A `'  _E  x  /\  x R B ) )
10 df-rex 2676 . 2  |-  ( E. x  e.  A  x R B  <->  E. x
( x  e.  A  /\  x R B ) )
117, 9, 103bitr4i 269 1  |-  ( A ( R  o.  `'  _E  ) B  <->  E. x  e.  A  x R B )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547    e. wcel 1721   E.wrex 2671   _Vcvv 2920   class class class wbr 4176    _E cep 4456   `'ccnv 4840    o. ccom 4845
This theorem is referenced by:  elfuns  25672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-eprel 4458  df-cnv 4849  df-co 4850
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