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Theorem dfcoss4 35816
 Description: Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 35812). (Contributed by Peter Mazsa, 12-Jul-2021.)
Assertion
Ref Expression
dfcoss4 𝑅 = ran (𝑅𝑅)

Proof of Theorem dfcoss4
Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-coss 35812 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
2 rnxrn 35799 . 2 ran (𝑅𝑅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
31, 2eqtr4i 2827 1 𝑅 = ran (𝑅𝑅)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538  ∃wex 1781   class class class wbr 5033  {copab 5095  ran crn 5524   ⋉ cxrn 35605   ≀ ccoss 35606 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336  df-1st 7675  df-2nd 7676  df-ec 8278  df-xrn 35776  df-coss 35812 This theorem is referenced by: (None)
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