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Theorem dfec2 7914
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 7913 . 2 [𝐴]𝑅 = (𝑅 “ {𝐴})
2 imasng 5645 . 2 (𝐴𝑉 → (𝑅 “ {𝐴}) = {𝑦𝐴𝑅𝑦})
31, 2syl5eq 2806 1 (𝐴𝑉 → [𝐴]𝑅 = {𝑦𝐴𝑅𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  {cab 2746  {csn 4321   class class class wbr 4804  cima 5269  [cec 7909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ec 7913
This theorem is referenced by:  elqsecl  7968  eqglact  17846  tgpconncompeqg  22116  fvline  32557  ellines  32565  ecres2  34368
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