Theorem dfcoss4 38379

Description: Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 38375). (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 38375	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2		rnxrn 38357	. 2 ⊢ ran (𝑅 ⋉ 𝑅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
3	1, 2	eqtr4i 2755	1 ⊢ ≀ 𝑅 = ran (𝑅 ⋉ 𝑅)

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   class class class wbr 5102  {copab 5164  ran crn 5632   ⋉ cxrn 38141   ≀ ccoss 38142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fo 6505  df-fv 6507  df-1st 7947  df-2nd 7948  df-ec 8650  df-xrn 38326  df-coss 38375

This theorem is referenced by: (None)