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Theorem relcoss 38405
Description: Cosets by 𝑅 is a relation. (Contributed by Peter Mazsa, 27-Dec-2018.)
Assertion
Ref Expression
relcoss Rel ≀ 𝑅

Proof of Theorem relcoss
Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-coss 38393 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
21relopabiv 5833 1 Rel ≀ 𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 395  wex 1776   class class class wbr 5148  Rel wrel 5694  ccoss 38162
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-opab 5211  df-xp 5695  df-rel 5696  df-coss 38393
This theorem is referenced by:  relcoels  38406  cocossss  38418  cnvcosseq  38419  refrelcoss3  38445  symrelcoss3  38447  1cosscnvxrn  38457  eleccossin  38465  cosselrels  38478  cnvrefrelcoss2  38519  eqvrelcoss3  38600
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