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Theorem elimasng 6088
Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) TODO: replace existing usages by usages of elimasng1 6086, remove, and relabel elimasng1 6086 to "elimasng".
Assertion
Ref Expression
elimasng ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))

Proof of Theorem elimasng
StepHypRef Expression
1 elimasng1 6086 . 2 ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
2 df-br 5150 . 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 205  wa 397  wcel 2107  {csn 4629  cop 4635   class class class wbr 5149  cima 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690
This theorem is referenced by:  elimasn  6089  inimasn  6156  dffv3  6888  fvimacnv  7055  fvrnressn  7159  elecg  8746  imasnopn  23194  imasncld  23195  imasncls  23196  ustelimasn  23727  blval2  24071  elbl4  24072  scutval  27301  iunsnima2  31848  1stpreimas  31927  opelco3  34746  funpartfv  34917  eltail  35259  elecALTV  37134  brtrclfv2  42478  frege77d  42497  dfafv23  45961
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