Theorem relcoss 38421

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 38409	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2	1	relopabiv 5786	1 ⊢ Rel ≀ 𝑅

Syntax hints:   ∧ wa 395  ∃wex 1779   class class class wbr 5110  Rel wrel 5646   ≀ ccoss 38176

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 3934  df-opab 5173  df-xp 5647  df-rel 5648  df-coss 38409

This theorem is referenced by:  relcoels  38422  cocossss  38434  cnvcosseq  38435  refrelcoss3  38461  symrelcoss3  38463  1cosscnvxrn  38473  eleccossin  38481  cosselrels  38494  cnvrefrelcoss2  38535  eqvrelcoss3  38616