Theorem relcoss 36158

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 36149	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2	1	relopabiv 5658	1 ⊢ Rel ≀ 𝑅

Syntax hints:   ∧ wa 399  ∃wex 1786   class class class wbr 5027  Rel wrel 5524   ≀ ccoss 35945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-v 3399  df-in 3848  df-ss 3858  df-opab 5090  df-xp 5525  df-rel 5526  df-coss 36149

This theorem is referenced by:  relcoels  36159  cocossss  36171  cnvcosseq  36172  refrelcoss3  36193  symrelcoss3  36195  1cosscnvxrn  36205  eleccossin  36213  cosselrels  36226  cnvrefrelcoss2  36263  eqvrelcoss3  36343