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

Theorem opelidres 5619
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 5479 . . 3 (𝐴𝑉𝐴 I 𝐴)
2 df-br 4844 . . 3 (𝐴 I 𝐴 ↔ ⟨𝐴, 𝐴⟩ ∈ I )
31, 2sylib 210 . 2 (𝐴𝑉 → ⟨𝐴, 𝐴⟩ ∈ I )
4 opelres 5606 . 2 (𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ (𝐴𝐵 ∧ ⟨𝐴, 𝐴⟩ ∈ I )))
53, 4mpbiran2d 700 1 (𝐴𝑉 → (⟨𝐴, 𝐴⟩ ∈ ( I ↾ 𝐵) ↔ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wcel 2157  cop 4374   class class class wbr 4843   I cid 5219  cres 5314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-res 5324
This theorem is referenced by:  idrefOLD  5727  ustfilxp  22344  ustelimasn  22354  metustfbas  22690  dfpo2  32159
  Copyright terms: Public domain W3C validator