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Theorem opelresiOLD 4982
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 4853 . . . 4  |-  ( A  e.  V  ->  A  _I  A )
2 df-br 4040 . . . 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 4978 . 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 1696   <.cop 3656   class class class wbr 4039    _I cid 4320    |` cres 4707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-res 4717
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