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Theorem dfec2 8748
Description: Alternate definition of 𝑅-coset of 𝐴. Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
dfec2 (𝐴𝑉 → [𝐴]𝑅 = {𝑦𝐴𝑅𝑦})
Distinct variable groups:   𝑦,𝐴   𝑦,𝑅
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem dfec2
StepHypRef Expression
1 df-ec 8747 . 2 [𝐴]𝑅 = (𝑅 “ {𝐴})
2 imasng 6102 . 2 (𝐴𝑉 → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
31, 2eqtrid 2789 1 (𝐴𝑉 → [𝐴]𝑅 = {𝑦𝐴𝑅𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {cab 2714  {csn 4626   class class class wbr 5143  cima 5688  [cec 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747
This theorem is referenced by:  elqsecl  8811  eqglact  19197  tgpconncompeqg  24120  fvline  36145  ellines  36153  ecres2  38280
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