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Theorem opelresi 5283
Description:  <. A ,  A >. belongs to a restriction of the identity class iff  A belongs to the restricting class. (Contributed by FL, 27-Oct-2008.) (Revised by NM, 30-Mar-2016.)
Assertion
Ref Expression
opelresi  |-  ( A  e.  V  ->  ( <. A ,  A >.  e.  (  _I  |`  B )  <-> 
A  e.  B ) )

Proof of Theorem opelresi
StepHypRef Expression
1 opelresg 5279 . 2  |-  ( A  e.  V  ->  ( <. A ,  A >.  e.  (  _I  |`  B )  <-> 
( <. A ,  A >.  e.  _I  /\  A  e.  B ) ) )
2 ididg 5154 . . . 4  |-  ( A  e.  V  ->  A  _I  A )
3 df-br 4448 . . . 4  |-  ( A  _I  A  <->  <. A ,  A >.  e.  _I  )
42, 3sylib 196 . . 3  |-  ( A  e.  V  ->  <. A ,  A >.  e.  _I  )
54biantrurd 508 . 2  |-  ( A  e.  V  ->  ( A  e.  B  <->  ( <. A ,  A >.  e.  _I  /\  A  e.  B
) ) )
61, 5bitr4d 256 1  |-  ( A  e.  V  ->  ( <. A ,  A >.  e.  (  _I  |`  B )  <-> 
A  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767   <.cop 4033   class class class wbr 4447    _I cid 4790    |` cres 5001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-res 5011
This theorem is referenced by:  issref  5378  ustfilxp  20450  ustelimasn  20460  metustfbasOLD  20803  metustfbas  20804
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