Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  opelidres Structured version   Visualization version   GIF version

Theorem opelidres 5839
 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
opelidres (𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴𝐵))

Proof of Theorem opelidres
StepHypRef Expression
1 ididg 5698 . . 3 (𝐴𝑉𝐴 I 𝐴)
2 df-br 5041 . . 3 (𝐴 I 𝐴 ↔ ⟨𝐴, 𝐴⟩ ∈ I )
31, 2sylib 220 . 2 (𝐴𝑉 → ⟨𝐴, 𝐴⟩ ∈ I )
4 opelres 5833 . 2 (𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ (𝐴𝐵 ∧ ⟨𝐴, 𝐴⟩ ∈ I )))
53, 4mpbiran2d 706 1 (𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2114  ⟨cop 4547   class class class wbr 5040   I cid 5433   ↾ cres 5531 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pr 5304 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-sn 4542  df-pr 4544  df-op 4548  df-br 5041  df-opab 5103  df-id 5434  df-xp 5535  df-rel 5536  df-res 5541 This theorem is referenced by:  ustfilxp  22794  ustelimasn  22804  metustfbas  23140  dfpo2  32996
 Copyright terms: Public domain W3C validator