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Theorem elimasng 6109
Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) TODO: replace existing usages by usages of elimasng1 6107, remove, and relabel elimasng1 6107 to "elimasng".
Assertion
Ref Expression
elimasng ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))

Proof of Theorem elimasng
StepHypRef Expression
1 elimasng1 6107 . 2 ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
2 df-br 5149 . 2 (𝐵𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
31, 2bitrdi 287 1 ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  {csn 4631  cop 4637   class class class wbr 5148  cima 5692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702
This theorem is referenced by:  elimasn  6110  inimasn  6178  dffv3  6903  fvimacnv  7073  fvrnressn  7181  elecg  8788  imasnopn  23714  imasncld  23715  imasncls  23716  ustelimasn  24247  blval2  24591  elbl4  24592  scutval  27860  iunsnima2  32639  1stpreimas  32721  opelco3  35756  funpartfv  35927  eltail  36357  elecALTV  38248  brtrclfv2  43717  frege77d  43736  dfafv23  47203
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