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Theorem relcoss 36546
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 36537 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
21relopabiv 5730 1 Rel ≀ 𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 396  wex 1782   class class class wbr 5074  Rel wrel 5594  ccoss 36333
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-opab 5137  df-xp 5595  df-rel 5596  df-coss 36537
This theorem is referenced by:  relcoels  36547  cocossss  36559  cnvcosseq  36560  refrelcoss3  36581  symrelcoss3  36583  1cosscnvxrn  36593  eleccossin  36601  cosselrels  36614  cnvrefrelcoss2  36651  eqvrelcoss3  36731
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