Theorem relcoss 38548

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 38536	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2	1	relopabiv 5766	1 ⊢ Rel ≀ 𝑅

Syntax hints:   ∧ wa 395  ∃wex 1780   class class class wbr 5095  Rel wrel 5626   ≀ ccoss 38245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-ss 3915  df-opab 5158  df-xp 5627  df-rel 5628  df-coss 38536

This theorem is referenced by:  relcoels  38549  cocossss  38561  cnvcosseq  38562  refrelcoss3  38588  symrelcoss3  38590  1cosscnvxrn  38600  eleccossin  38608  cosselrels  38610  cnvrefrelcoss2  38652  eqvrelcoss3  38737