Theorem dfcoss4 39439

Description: Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 39435). (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 39435	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2		rnxrn 39355	. 2 ⊢ ran (𝑅 ⋉ 𝑅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
3	1, 2	eqtr4i 2787	1 ⊢ ≀ 𝑅 = ran (𝑅 ⋉ 𝑅)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   class class class wbr 5103  {copab 5167  ran crn 5652   ⋉ cxrn 39108   ≀ ccoss 39117

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3900  df-un 3902  df-in 3904  df-ss 3915  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-fo 6545  df-fv 6547  df-1st 8002  df-2nd 8003  df-ec 8720  df-xrn 39314  df-coss 39435

This theorem is used by: (None)