Theorem relcoss 36958

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 36946	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2	1	relopabiv 5781	1 ⊢ Rel ≀ 𝑅

Syntax hints:   ∧ wa 396  ∃wex 1781   class class class wbr 5110  Rel wrel 5643   ≀ ccoss 36707

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3448  df-in 3920  df-ss 3930  df-opab 5173  df-xp 5644  df-rel 5645  df-coss 36946

This theorem is referenced by:  relcoels  36959  cocossss  36971  cnvcosseq  36972  refrelcoss3  36998  symrelcoss3  37000  1cosscnvxrn  37010  eleccossin  37018  cosselrels  37031  cnvrefrelcoss2  37072  eqvrelcoss3  37153