Theorem relcoss 37596

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 37584	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2	1	relopabiv 5819	1 ⊢ Rel ≀ 𝑅

Syntax hints:   ∧ wa 394  ∃wex 1779   class class class wbr 5147  Rel wrel 5680   ≀ ccoss 37346

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-in 3954  df-ss 3964  df-opab 5210  df-xp 5681  df-rel 5682  df-coss 37584

This theorem is referenced by:  relcoels  37597  cocossss  37609  cnvcosseq  37610  refrelcoss3  37636  symrelcoss3  37638  1cosscnvxrn  37648  eleccossin  37656  cosselrels  37669  cnvrefrelcoss2  37710  eqvrelcoss3  37791