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Theorem opelresiOLD 4966
Description:  <. A ,  A >. belongs to a restriction of the identity class iff  A belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
opelresiOLD  |-  ( A  e.  V  ->  ( A  e.  B  <->  <. A ,  A >.  e.  (  _I  |`  B ) ) )

Proof of Theorem opelresiOLD
StepHypRef Expression
1 ididg 4837 . . . 4  |-  ( A  e.  V  ->  A  _I  A )
2 df-br 4024 . . . 4  |-  ( A  _I  A  <->  <. A ,  A >.  e.  _I  )
31, 2sylib 188 . . 3  |-  ( A  e.  V  ->  <. A ,  A >.  e.  _I  )
43biantrurd 494 . 2  |-  ( A  e.  V  ->  ( A  e.  B  <->  ( <. A ,  A >.  e.  _I  /\  A  e.  B
) ) )
5 opelresg 4962 . 2  |-  ( A  e.  V  ->  ( <. A ,  A >.  e.  (  _I  |`  B )  <-> 
( <. A ,  A >.  e.  _I  /\  A  e.  B ) ) )
64, 5bitr4d 247 1  |-  ( A  e.  V  ->  ( A  e.  B  <->  <. A ,  A >.  e.  (  _I  |`  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   <.cop 3643   class class class wbr 4023    _I cid 4304    |` cres 4691
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-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-id 4309  df-xp 4695  df-rel 4696  df-res 4701
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