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Theorem opelresi 4900
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 (𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴𝐵))

Proof of Theorem opelresi
StepHypRef Expression
1 opelresg 4896 . 2 (𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ (⟨𝐴, 𝐴⟩ ∈ I ∧ 𝐴𝐵)))
2 ididg 4762 . . . 4 (𝐴𝑉𝐴 I 𝐴)
3 df-br 3988 . . . 4 (𝐴 I 𝐴 ↔ ⟨𝐴, 𝐴⟩ ∈ I )
42, 3sylib 121 . . 3 (𝐴𝑉 → ⟨𝐴, 𝐴⟩ ∈ I )
54biantrurd 303 . 2 (𝐴𝑉 → (𝐴𝐵 ↔ (⟨𝐴, 𝐴⟩ ∈ I ∧ 𝐴𝐵)))
61, 5bitr4d 190 1 (𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 2141  cop 3584   class class class wbr 3987   I cid 4271  cres 4611
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 4105  ax-pow 4158  ax-pr 4192
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 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-res 4621
This theorem is referenced by:  issref  4991
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