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Theorem coep 24108
Description: Composition with epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
Hypotheses
Ref Expression
coep.1  |-  A  e. 
_V
coep.2  |-  B  e. 
_V
Assertion
Ref Expression
coep  |-  ( A (  _E  o.  R
) B  <->  E. x  e.  B  A R x )
Distinct variable groups:    x, A    x, B    x, R

Proof of Theorem coep
StepHypRef Expression
1 coep.2 . . . . . 6  |-  B  e. 
_V
21epelc 4307 . . . . 5  |-  ( x  _E  B  <->  x  e.  B )
32anbi2i 675 . . . 4  |-  ( ( A R x  /\  x  _E  B )  <->  ( A R x  /\  x  e.  B )
)
4 ancom 437 . . . 4  |-  ( ( A R x  /\  x  e.  B )  <->  ( x  e.  B  /\  A R x ) )
53, 4bitri 240 . . 3  |-  ( ( A R x  /\  x  _E  B )  <->  ( x  e.  B  /\  A R x ) )
65exbii 1569 . 2  |-  ( E. x ( A R x  /\  x  _E  B )  <->  E. x
( x  e.  B  /\  A R x ) )
7 coep.1 . . 3  |-  A  e. 
_V
87, 1brco 4852 . 2  |-  ( A (  _E  o.  R
) B  <->  E. x
( A R x  /\  x  _E  B
) )
9 df-rex 2549 . 2  |-  ( E. x  e.  B  A R x  <->  E. x ( x  e.  B  /\  A R x ) )
106, 8, 93bitr4i 268 1  |-  ( A (  _E  o.  R
) B  <->  E. x  e.  B  A R x )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    e. wcel 1684   E.wrex 2544   _Vcvv 2788   class class class wbr 4023    _E cep 4303    o. ccom 4693
This theorem is referenced by:  dffr5  24110  brbigcup  24438  elfuns  24454  brimage  24465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-eprel 4305  df-co 4698
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