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Theorem predep 25018
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 24994 . 2  |-  Pred (  _E  ,  A ,  X
)  =  ( A  i^i  ( `'  _E  " { X } ) )
2 relcnv 5154 . . . . 5  |-  Rel  `'  _E
3 relimasn 5139 . . . . 5  |-  ( Rel  `'  _E  ->  ( `'  _E  " { X }
)  =  { y  |  X `'  _E  y } )
42, 3ax-mp 8 . . . 4  |-  ( `'  _E  " { X } )  =  {
y  |  X `'  _E  y }
5 vex 2876 . . . . . . 7  |-  y  e. 
_V
6 brcnvg 4965 . . . . . . 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 4409 . . . . . 6  |-  ( X  e.  B  ->  (
y  _E  X  <->  y  e.  X ) )
97, 8bitrd 244 . . . . 5  |-  ( X  e.  B  ->  ( X `'  _E  y  <->  y  e.  X ) )
109abbi1dv 2482 . . . 4  |-  ( X  e.  B  ->  { y  |  X `'  _E  y }  =  X
)
114, 10syl5eq 2410 . . 3  |-  ( X  e.  B  ->  ( `'  _E  " { X } )  =  X )
1211ineq2d 3458 . 2  |-  ( X  e.  B  ->  ( A  i^i  ( `'  _E  " { X } ) )  =  ( A  i^i  X ) )
131, 12syl5eq 2410 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 1647    e. wcel 1715   {cab 2352   _Vcvv 2873    i^i cin 3237   {csn 3729   class class class wbr 4125    _E cep 4406   `'ccnv 4791   "cima 4795   Rel wrel 4797   Predcpred 24993
This theorem is referenced by:  predon  25019  epsetlike  25020  omsinds  25045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-eprel 4408  df-xp 4798  df-rel 4799  df-cnv 4800  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-pred 24994
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