Theorem dfcoss4 36541

Description: Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 36537). (Contributed by Peter Mazsa, 12-Jul-2021.)

Assertion

Ref	Expression
dfcoss4	⊢ ≀ 𝑅 = ran (𝑅 ⋉ 𝑅)

Proof of Theorem dfcoss4

Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-coss 36537	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2		rnxrn 36524	. 2 ⊢ ran (𝑅 ⋉ 𝑅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
3	1, 2	eqtr4i 2769	1 ⊢ ≀ 𝑅 = ran (𝑅 ⋉ 𝑅)

Syntax hints:   ∧ wa 396   = wceq 1539  ∃wex 1782   class class class wbr 5074  {copab 5136  ran crn 5590   ⋉ cxrn 36332   ≀ ccoss 36333

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-1st 7831  df-2nd 7832  df-ec 8500  df-xrn 36501  df-coss 36537

This theorem is referenced by: (None)