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Theorem opelresi 5566
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 ididg 5431 . . 3 (𝐴𝑉𝐴 I 𝐴)
2 df-br 4805 . . 3 (𝐴 I 𝐴 ↔ ⟨𝐴, 𝐴⟩ ∈ I )
31, 2sylib 208 . 2 (𝐴𝑉 → ⟨𝐴, 𝐴⟩ ∈ I )
4 opelresg 5557 . 2 (𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ (⟨𝐴, 𝐴⟩ ∈ I ∧ 𝐴𝐵)))
53, 4mpbirand 531 1 (𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2139  cop 4327   class class class wbr 4804   I cid 5173  cres 5268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-res 5278
This theorem is referenced by:  issref  5667  ustfilxp  22217  ustelimasn  22227  metustfbas  22563
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