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Theorem predep 24192
Description: The predecessor under the epsilon relationship is equivalent to an intersection. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
predep  |-  ( X  e.  B  ->  Pred (  _E  ,  A ,  X
)  =  ( A  i^i  X ) )

Proof of Theorem predep
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-pred 24168 . 2  |-  Pred (  _E  ,  A ,  X
)  =  ( A  i^i  ( `'  _E  " { X } ) )
2 relcnv 5051 . . . . 5  |-  Rel  `'  _E
3 relimasn 5036 . . . . 5  |-  ( Rel  `'  _E  ->  ( `'  _E  " { X }
)  =  { y  |  X `'  _E  y } )
42, 3ax-mp 8 . . . 4  |-  ( `'  _E  " { X } )  =  {
y  |  X `'  _E  y }
5 vex 2791 . . . . . . 7  |-  y  e. 
_V
6 brcnvg 4862 . . . . . . 7  |-  ( ( X  e.  B  /\  y  e.  _V )  ->  ( X `'  _E  y 
<->  y  _E  X ) )
75, 6mpan2 652 . . . . . 6  |-  ( X  e.  B  ->  ( X `'  _E  y  <->  y  _E  X ) )
8 epelg 4306 . . . . . 6  |-  ( X  e.  B  ->  (
y  _E  X  <->  y  e.  X ) )
97, 8bitrd 244 . . . . 5  |-  ( X  e.  B  ->  ( X `'  _E  y  <->  y  e.  X ) )
109abbi1dv 2399 . . . 4  |-  ( X  e.  B  ->  { y  |  X `'  _E  y }  =  X
)
114, 10syl5eq 2327 . . 3  |-  ( X  e.  B  ->  ( `'  _E  " { X } )  =  X )
1211ineq2d 3370 . 2  |-  ( X  e.  B  ->  ( A  i^i  ( `'  _E  " { X } ) )  =  ( A  i^i  X ) )
131, 12syl5eq 2327 1  |-  ( X  e.  B  ->  Pred (  _E  ,  A ,  X
)  =  ( A  i^i  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788    i^i cin 3151   {csn 3640   class class class wbr 4023    _E cep 4303   `'ccnv 4688   "cima 4692   Rel wrel 4694   Predcpred 24167
This theorem is referenced by:  predon  24193  epsetlike  24194  omsinds  24219
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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-pred 24168
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