Theorem relcoss 38972

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 38960	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2	1	relopabiv 5789	1 ⊢ Rel ≀ 𝑅

Syntax hints:   ∧ wa 399  ∃wex 1798   class class class wbr 5097  Rel wrel 5648   ≀ ccoss 38642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-ss 3919  df-opab 5160  df-xp 5649  df-rel 5650  df-coss 38960

This theorem is referenced by:  relcoels  38973  cocossss  38985  cnvcosseq  38986  refrelcoss3  39012  symrelcoss3  39014  1cosscnvxrn  39024  eleccossin  39032  cosselrels  39034  cnvrefrelcoss2  39076  eqvrelcoss3  39161