Theorem relcoss 38848

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.

Step	Hyp	Ref	Expression
1		df-coss 38836	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2	1	relopabiv 5769	1 ⊢ Rel ≀ 𝑅

Syntax hints:   ∧ wa 395  ∃wex 1781   class class class wbr 5086  Rel wrel 5629   ≀ ccoss 38518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-ss 3907  df-opab 5149  df-xp 5630  df-rel 5631  df-coss 38836

This theorem is referenced by:  relcoels  38849  cocossss  38861  cnvcosseq  38862  refrelcoss3  38888  symrelcoss3  38890  1cosscnvxrn  38900  eleccossin  38908  cosselrels  38910  cnvrefrelcoss2  38952  eqvrelcoss3  39037