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Theorem opelresi 4539
 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 𝑉 → (⟨A, A ( I ↾ B) ↔ A B))

Proof of Theorem opelresi
StepHypRef Expression
1 opelresg 4535 . 2 (A 𝑉 → (⟨A, A ( I ↾ B) ↔ (⟨A, A I A B)))
2 ididg 4405 . . . 4 (A 𝑉A I A)
3 df-br 3729 . . . 4 (A I A ↔ ⟨A, A I )
42, 3sylib 127 . . 3 (A 𝑉 → ⟨A, A I )
54biantrurd 289 . 2 (A 𝑉 → (A B ↔ (⟨A, A I A B)))
61, 5bitr4d 180 1 (A 𝑉 → (⟨A, A ( I ↾ B) ↔ A B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1367  ⟨cop 3343   class class class wbr 3728   I cid 3989   ↾ cres 4263 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-sep 3839  ax-pow 3891  ax-pr 3908 This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-nf 1324  df-sb 1620  df-eu 1877  df-mo 1878  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-ral 2281  df-rex 2282  df-v 2529  df-un 2891  df-in 2893  df-ss 2900  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-br 3729  df-opab 3783  df-id 3994  df-xp 4267  df-rel 4268  df-res 4273 This theorem is referenced by:  issref  4623
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