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Theorem predep 25414
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 25390 . 2  |-  Pred (  _E  ,  A ,  X
)  =  ( A  i^i  ( `'  _E  " { X } ) )
2 relcnv 5209 . . . . 5  |-  Rel  `'  _E
3 relimasn 5194 . . . . 5  |-  ( Rel  `'  _E  ->  ( `'  _E  " { X }
)  =  { y  |  X `'  _E  y } )
42, 3ax-mp 8 . . . 4  |-  ( `'  _E  " { X } )  =  {
y  |  X `'  _E  y }
5 vex 2927 . . . . . . 7  |-  y  e. 
_V
6 brcnvg 5020 . . . . . . 7  |-  ( ( X  e.  B  /\  y  e.  _V )  ->  ( X `'  _E  y 
<->  y  _E  X ) )
75, 6mpan2 653 . . . . . 6  |-  ( X  e.  B  ->  ( X `'  _E  y  <->  y  _E  X ) )
8 epelg 4463 . . . . . 6  |-  ( X  e.  B  ->  (
y  _E  X  <->  y  e.  X ) )
97, 8bitrd 245 . . . . 5  |-  ( X  e.  B  ->  ( X `'  _E  y  <->  y  e.  X ) )
109abbi1dv 2528 . . . 4  |-  ( X  e.  B  ->  { y  |  X `'  _E  y }  =  X
)
114, 10syl5eq 2456 . . 3  |-  ( X  e.  B  ->  ( `'  _E  " { X } )  =  X )
1211ineq2d 3510 . 2  |-  ( X  e.  B  ->  ( A  i^i  ( `'  _E  " { X } ) )  =  ( A  i^i  X ) )
131, 12syl5eq 2456 1  |-  ( X  e.  B  ->  Pred (  _E  ,  A ,  X
)  =  ( A  i^i  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   {cab 2398   _Vcvv 2924    i^i cin 3287   {csn 3782   class class class wbr 4180    _E cep 4460   `'ccnv 4844   "cima 4848   Rel wrel 4850   Predcpred 25389
This theorem is referenced by:  predon  25415  epsetlike  25416  omsinds  25441
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-eprel 4462  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 25390
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