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Theorem opelresi 4933
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 4929 . 2 (𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ (⟨𝐴, 𝐴⟩ ∈ I ∧ 𝐴𝐵)))
2 ididg 4795 . . . 4 (𝐴𝑉𝐴 I 𝐴)
3 df-br 4019 . . . 4 (𝐴 I 𝐴 ↔ ⟨𝐴, 𝐴⟩ ∈ I )
42, 3sylib 122 . . 3 (𝐴𝑉 → ⟨𝐴, 𝐴⟩ ∈ I )
54biantrurd 305 . 2 (𝐴𝑉 → (𝐴𝐵 ↔ (⟨𝐴, 𝐴⟩ ∈ I ∧ 𝐴𝐵)))
61, 5bitr4d 191 1 (𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wcel 2160  cop 3610   class class class wbr 4018   I cid 4303  cres 4643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-res 4653
This theorem is referenced by:  issref  5026
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