Theorem relcoss 38409

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 38397	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2	1	relopabiv 5785	1 ⊢ Rel ≀ 𝑅

Syntax hints:   ∧ wa 395  ∃wex 1779   class class class wbr 5109  Rel wrel 5645   ≀ ccoss 38164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-ss 3933  df-opab 5172  df-xp 5646  df-rel 5647  df-coss 38397

This theorem is referenced by:  relcoels  38410  cocossss  38422  cnvcosseq  38423  refrelcoss3  38449  symrelcoss3  38451  1cosscnvxrn  38461  eleccossin  38469  cosselrels  38482  cnvrefrelcoss2  38523  eqvrelcoss3  38604