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Theorem opelresi 4712
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 4708 . 2  |-  ( A  e.  V  ->  ( <. A ,  A >.  e.  (  _I  |`  B )  <-> 
( <. A ,  A >.  e.  _I  /\  A  e.  B ) ) )
2 ididg 4577 . . . 4  |-  ( A  e.  V  ->  A  _I  A )
3 df-br 3838 . . . 4  |-  ( A  _I  A  <->  <. A ,  A >.  e.  _I  )
42, 3sylib 120 . . 3  |-  ( A  e.  V  ->  <. A ,  A >.  e.  _I  )
54biantrurd 299 . 2  |-  ( A  e.  V  ->  ( A  e.  B  <->  ( <. A ,  A >.  e.  _I  /\  A  e.  B
) ) )
61, 5bitr4d 189 1  |-  ( A  e.  V  ->  ( <. A ,  A >.  e.  (  _I  |`  B )  <-> 
A  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1438   <.cop 3444   class class class wbr 3837    _I cid 4106    |` cres 4430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-res 4440
This theorem is referenced by:  issref  4801
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