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Theorem predep 25472
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 25444 . 2  |-  Pred (  _E  ,  A ,  X
)  =  ( A  i^i  ( `'  _E  " { X } ) )
2 relcnv 5245 . . . . 5  |-  Rel  `'  _E
3 relimasn 5230 . . . . 5  |-  ( Rel  `'  _E  ->  ( `'  _E  " { X }
)  =  { y  |  X `'  _E  y } )
42, 3ax-mp 5 . . . 4  |-  ( `'  _E  " { X } )  =  {
y  |  X `'  _E  y }
5 vex 2961 . . . . . . 7  |-  y  e. 
_V
6 brcnvg 5056 . . . . . . 7  |-  ( ( X  e.  B  /\  y  e.  _V )  ->  ( X `'  _E  y 
<->  y  _E  X ) )
75, 6mpan2 654 . . . . . 6  |-  ( X  e.  B  ->  ( X `'  _E  y  <->  y  _E  X ) )
8 epelg 4498 . . . . . 6  |-  ( X  e.  B  ->  (
y  _E  X  <->  y  e.  X ) )
97, 8bitrd 246 . . . . 5  |-  ( X  e.  B  ->  ( X `'  _E  y  <->  y  e.  X ) )
109abbi1dv 2554 . . . 4  |-  ( X  e.  B  ->  { y  |  X `'  _E  y }  =  X
)
114, 10syl5eq 2482 . . 3  |-  ( X  e.  B  ->  ( `'  _E  " { X } )  =  X )
1211ineq2d 3544 . 2  |-  ( X  e.  B  ->  ( A  i^i  ( `'  _E  " { X } ) )  =  ( A  i^i  X ) )
131, 12syl5eq 2482 1  |-  ( X  e.  B  ->  Pred (  _E  ,  A ,  X
)  =  ( A  i^i  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   {cab 2424   _Vcvv 2958    i^i cin 3321   {csn 3816   class class class wbr 4215    _E cep 4495   `'ccnv 4880   "cima 4884   Rel wrel 4886   Predcpred 25443
This theorem is referenced by:  predon  25473  epsetlike  25474  omsinds  25499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-eprel 4497  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-pred 25444
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