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Theorem opelresi 4902
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 4898 . 2  |-  ( A  e.  V  ->  ( <. A ,  A >.  e.  (  _I  |`  B )  <-> 
( <. A ,  A >.  e.  _I  /\  A  e.  B ) ) )
2 ididg 4764 . . . 4  |-  ( A  e.  V  ->  A  _I  A )
3 df-br 3990 . . . 4  |-  ( A  _I  A  <->  <. A ,  A >.  e.  _I  )
42, 3sylib 121 . . 3  |-  ( A  e.  V  ->  <. A ,  A >.  e.  _I  )
54biantrurd 303 . 2  |-  ( A  e.  V  ->  ( A  e.  B  <->  ( <. A ,  A >.  e.  _I  /\  A  e.  B
) ) )
61, 5bitr4d 190 1  |-  ( A  e.  V  ->  ( <. A ,  A >.  e.  (  _I  |`  B )  <-> 
A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 2141   <.cop 3586   class class class wbr 3989    _I cid 4273    |` cres 4613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-res 4623
This theorem is referenced by:  issref  4993
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