Theorem relcoss 39222

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 39210	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2	1	relopabiv 5809	1 ⊢ Rel ≀ 𝑅

This proof depends on syntax axioms:   ∧ wa 401  ∃wex 1812   class class class wbr 5111  Rel wrel 5668   ≀ ccoss 38892

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-ss 3923  df-opab 5176  df-xp 5669  df-rel 5670  df-coss 39210

This theorem is used by:  relcoels  39223  cocossss  39235  cnvcosseq  39236  refrelcoss3  39262  symrelcoss3  39264  1cosscnvxrn  39274  eleccossin  39282  cosselrels  39284  cnvrefrelcoss2  39326  eqvrelcoss3  39411