Theorem relcoss 37231

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 37219	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2	1	relopabiv 5818	1 ⊢ Rel ≀ 𝑅

Syntax hints:   ∧ wa 397  ∃wex 1782   class class class wbr 5147  Rel wrel 5680   ≀ ccoss 36981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3954  df-ss 3964  df-opab 5210  df-xp 5681  df-rel 5682  df-coss 37219

This theorem is referenced by:  relcoels  37232  cocossss  37244  cnvcosseq  37245  refrelcoss3  37271  symrelcoss3  37273  1cosscnvxrn  37283  eleccossin  37291  cosselrels  37304  cnvrefrelcoss2  37345  eqvrelcoss3  37426