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Theorem opelresi 4830
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 4826 . 2 (𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ (⟨𝐴, 𝐴⟩ ∈ I ∧ 𝐴𝐵)))
2 ididg 4692 . . . 4 (𝐴𝑉𝐴 I 𝐴)
3 df-br 3930 . . . 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 1480  cop 3530   class class class wbr 3929   I cid 4210  cres 4541
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-res 4551
This theorem is referenced by:  issref  4921
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