Theorem relcoss 39582

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 39570	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2	1	relopabiv 5799	1 ⊢ Rel ≀ 𝑅

This proof depends on syntax axioms:   ∧ wa 401  ∃wex 1811   class class class wbr 5102  Rel wrel 5656   ≀ ccoss 39252

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-opab 5167  df-xp 5657  df-rel 5658  df-coss 39570

This theorem is used by:  relcoels  39583  cocossss  39595  cnvcosseq  39596  refrelcoss3  39622  symrelcoss3  39624  1cosscnvxrn  39634  eleccossin  39642  cosselrels  39644  cnvrefrelcoss2  39686  eqvrelcoss3  39771