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Theorem opelresi 4651
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 4647 . 2 (𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ (⟨𝐴, 𝐴⟩ ∈ I ∧ 𝐴𝐵)))
2 ididg 4517 . . . 4 (𝐴𝑉𝐴 I 𝐴)
3 df-br 3793 . . . 4 (𝐴 I 𝐴 ↔ ⟨𝐴, 𝐴⟩ ∈ I )
42, 3sylib 131 . . 3 (𝐴𝑉 → ⟨𝐴, 𝐴⟩ ∈ I )
54biantrurd 293 . 2 (𝐴𝑉 → (𝐴𝐵 ↔ (⟨𝐴, 𝐴⟩ ∈ I ∧ 𝐴𝐵)))
61, 5bitr4d 184 1 (𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wcel 1409  cop 3406   class class class wbr 3792   I cid 4053  cres 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-res 4385
This theorem is referenced by:  issref  4735
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