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Theorem relrngo 36405
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 β„Ž π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rngo 36404 . 2 RingOps = {βŸ¨π‘”, β„ŽβŸ© ∣ ((𝑔 ∈ AbelOp ∧ β„Ž:(ran 𝑔 Γ— ran 𝑔)⟢ran 𝑔) ∧ (βˆ€π‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran π‘”βˆ€π‘§ ∈ ran 𝑔(((π‘₯β„Žπ‘¦)β„Žπ‘§) = (π‘₯β„Ž(π‘¦β„Žπ‘§)) ∧ (π‘₯β„Ž(𝑦𝑔𝑧)) = ((π‘₯β„Žπ‘¦)𝑔(π‘₯β„Žπ‘§)) ∧ ((π‘₯𝑔𝑦)β„Žπ‘§) = ((π‘₯β„Žπ‘§)𝑔(π‘¦β„Žπ‘§))) ∧ βˆƒπ‘₯ ∈ ran π‘”βˆ€π‘¦ ∈ ran 𝑔((π‘₯β„Žπ‘¦) = 𝑦 ∧ (π‘¦β„Žπ‘₯) = 𝑦)))}
21relopabiv 5780 1 Rel RingOps
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070   Γ— cxp 5635  ran crn 5638  Rel wrel 5642  βŸΆwf 6496  (class class class)co 7361  AbelOpcablo 29535  RingOpscrngo 36403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-in 3921  df-ss 3931  df-opab 5172  df-xp 5643  df-rel 5644  df-rngo 36404
This theorem is referenced by:  isrngo  36406  rngoi  36408  rngoablo2  36418  rngosn3  36433  isdrngo1  36465  iscrngo2  36506
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