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Theorem coepr 25380
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 2961 . . . . . 6  |-  x  e. 
_V
31, 2brcnv 5058 . . . . 5  |-  ( A `'  _E  x  <->  x  _E  A )
41epelc 4499 . . . . 5  |-  ( x  _E  A  <->  x  e.  A )
53, 4bitri 242 . . . 4  |-  ( A `'  _E  x  <->  x  e.  A )
65anbi1i 678 . . 3  |-  ( ( A `'  _E  x  /\  x R B )  <-> 
( x  e.  A  /\  x R B ) )
76exbii 1593 . 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 5046 . 2  |-  ( A ( R  o.  `'  _E  ) B  <->  E. x
( A `'  _E  x  /\  x R B ) )
10 df-rex 2713 . 2  |-  ( E. x  e.  A  x R B  <->  E. x
( x  e.  A  /\  x R B ) )
117, 9, 103bitr4i 270 1  |-  ( A ( R  o.  `'  _E  ) B  <->  E. x  e.  A  x R B )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1551    e. wcel 1726   E.wrex 2708   _Vcvv 2958   class class class wbr 4215    _E cep 4495   `'ccnv 4880    o. ccom 4885
This theorem is referenced by:  elfuns  25765  brub  25804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pr 4406
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-rex 2713  df-rab 2716  df-v 2960  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 4216  df-opab 4270  df-eprel 4497  df-cnv 4889  df-co 4890
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