Theorem dfcoss4 36227

Description: Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 36223). (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 36223	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2		rnxrn 36210	. 2 ⊢ ran (𝑅 ⋉ 𝑅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
3	1, 2	eqtr4i 2762	1 ⊢ ≀ 𝑅 = ran (𝑅 ⋉ 𝑅)

Syntax hints:   ∧ wa 399   = wceq 1543  ∃wex 1787   class class class wbr 5039  {copab 5101  ran crn 5537   ⋉ cxrn 36018   ≀ ccoss 36019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-fo 6364  df-fv 6366  df-1st 7739  df-2nd 7740  df-ec 8371  df-xrn 36187  df-coss 36223

This theorem is referenced by: (None)