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Theorem coep 25366
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 4488 . . . . 5  |-  ( x  _E  B  <->  x  e.  B )
32anbi2i 676 . . . 4  |-  ( ( A R x  /\  x  _E  B )  <->  ( A R x  /\  x  e.  B )
)
4 ancom 438 . . . 4  |-  ( ( A R x  /\  x  e.  B )  <->  ( x  e.  B  /\  A R x ) )
53, 4bitri 241 . . 3  |-  ( ( A R x  /\  x  _E  B )  <->  ( x  e.  B  /\  A R x ) )
65exbii 1592 . 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 5035 . 2  |-  ( A (  _E  o.  R
) B  <->  E. x
( A R x  /\  x  _E  B
) )
9 df-rex 2703 . 2  |-  ( E. x  e.  B  A R x  <->  E. x ( x  e.  B  /\  A R x ) )
106, 8, 93bitr4i 269 1  |-  ( A (  _E  o.  R
) B  <->  E. x  e.  B  A R x )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    e. wcel 1725   E.wrex 2698   _Vcvv 2948   class class class wbr 4204    _E cep 4484    o. ccom 4874
This theorem is referenced by:  dffr5  25368  brbigcup  25735  elfuns  25752  brimage  25763
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-eprel 4486  df-co 4879
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