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

Theorem elimasng 6038
Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) TODO: replace existing usages by usages of elimasng1 6036, remove, and relabel elimasng1 6036 to "elimasng".
Assertion
Ref Expression
elimasng ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))

Proof of Theorem elimasng
StepHypRef Expression
1 elimasng1 6036 . 2 ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
2 df-br 5104 . 2 (𝐵𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
31, 2bitrdi 286 1 ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  {csn 4584  cop 4590   class class class wbr 5103  cima 5634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644
This theorem is referenced by:  elimasn  6039  inimasn  6106  dffv3  6835  fvimacnv  7000  fvrnressn  7103  elecg  8649  imasnopn  22987  imasncld  22988  imasncls  22989  ustelimasn  23520  blval2  23864  elbl4  23865  scutval  27085  iunsnima2  31383  1stpreimas  31462  opelco3  34181  funpartfv  34462  eltail  34778  elecALTV  36658  brtrclfv2  41904  frege77d  41923  dfafv23  45380
  Copyright terms: Public domain W3C validator