Theorem relcoss 38424

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 38412	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2	1	relopabiv 5830	1 ⊢ Rel ≀ 𝑅

Syntax hints:   ∧ wa 395  ∃wex 1779   class class class wbr 5143  Rel wrel 5690   ≀ ccoss 38182

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-opab 5206  df-xp 5691  df-rel 5692  df-coss 38412

This theorem is referenced by:  relcoels  38425  cocossss  38437  cnvcosseq  38438  refrelcoss3  38464  symrelcoss3  38466  1cosscnvxrn  38476  eleccossin  38484  cosselrels  38497  cnvrefrelcoss2  38538  eqvrelcoss3  38619