Theorem relcoss 38960

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 38948	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2	1	relopabiv 5786	1 ⊢ Rel ≀ 𝑅

Syntax hints:   ∧ wa 398  ∃wex 1793   class class class wbr 5094  Rel wrel 5645   ≀ ccoss 38630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1557  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-v 3450  df-ss 3916  df-opab 5157  df-xp 5646  df-rel 5647  df-coss 38948

This theorem is referenced by:  relcoels  38961  cocossss  38973  cnvcosseq  38974  refrelcoss3  39000  symrelcoss3  39002  1cosscnvxrn  39012  eleccossin  39020  cosselrels  39022  cnvrefrelcoss2  39064  eqvrelcoss3  39149