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Theorem coep 25134
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 4439 . . . . 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 1589 . 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 4985 . 2  |-  ( A (  _E  o.  R
) B  <->  E. x
( A R x  /\  x  _E  B
) )
9 df-rex 2657 . 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 1547    e. wcel 1717   E.wrex 2652   _Vcvv 2901   class class class wbr 4155    _E cep 4435    o. ccom 4824
This theorem is referenced by:  dffr5  25136  brbigcup  25464  elfuns  25480  brimage  25491
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-eprel 4437  df-co 4829
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