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Theorem relrngo 37805
Description: The class of all unital rings is a relation. (Contributed by FL, 31-Aug-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
relrngo Rel RingOps

Proof of Theorem relrngo
Dummy variables 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rngo 37804 . 2 RingOps = {⟨𝑔, ⟩ ∣ ((𝑔 ∈ AbelOp ∧ :(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔𝑧 ∈ ran 𝑔(((𝑥𝑦)𝑧) = (𝑥(𝑦𝑧)) ∧ (𝑥(𝑦𝑔𝑧)) = ((𝑥𝑦)𝑔(𝑥𝑧)) ∧ ((𝑥𝑔𝑦)𝑧) = ((𝑥𝑧)𝑔(𝑦𝑧))) ∧ ∃𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔((𝑥𝑦) = 𝑦 ∧ (𝑦𝑥) = 𝑦)))}
21relopabiv 5843 1 Rel RingOps
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3a 1087   = wceq 1537  wcel 2103  wral 3063  wrex 3072   × cxp 5697  ran crn 5700  Rel wrel 5704  wf 6568  (class class class)co 7445  AbelOpcablo 30567  RingOpscrngo 37803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3484  df-ss 3987  df-opab 5232  df-xp 5705  df-rel 5706  df-rngo 37804
This theorem is referenced by:  isrngo  37806  rngoi  37808  rngoablo2  37818  rngosn3  37833  isdrngo1  37865  iscrngo2  37906
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