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Theorem dfec2 8694
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 8693 . 2 [𝐴]𝑅 = (𝑅 “ {𝐴})
2 imasng 6074 . 2 (𝐴𝑉 → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
31, 2eqtrid 2785 1 (𝐴𝑉 → [𝐴]𝑅 = {𝑦𝐴𝑅𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {cab 2710  {csn 4624   class class class wbr 5144  cima 5675  [cec 8689
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 5295  ax-nul 5302  ax-pr 5423
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 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ec 8693
This theorem is referenced by:  elqsecl  8753  eqglact  19044  tgpconncompeqg  23585  fvline  35047  ellines  35055  ecres2  37053
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