Theorem relcoss 39047

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 39035	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2	1	relopabiv 5805	1 ⊢ Rel ≀ 𝑅

Syntax hints:   ∧ wa 400  ∃wex 1806   class class class wbr 5110  Rel wrel 5664   ≀ ccoss 38717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-ss 3930  df-opab 5175  df-xp 5665  df-rel 5666  df-coss 39035

This theorem is referenced by:  relcoels  39048  cocossss  39060  cnvcosseq  39061  refrelcoss3  39087  symrelcoss3  39089  1cosscnvxrn  39099  eleccossin  39107  cosselrels  39109  cnvrefrelcoss2  39151  eqvrelcoss3  39236