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Theorem reldvdsr 18467
 Description: The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypothesis
Ref Expression
reldvdsr.1 = (∥r𝑅)
Assertion
Ref Expression
reldvdsr Rel

Proof of Theorem reldvdsr
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dvdsr 18464 . . 3 r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r𝑤)𝑥) = 𝑦)})
21relmptopab 6781 . 2 Rel (∥r𝑅)
3 reldvdsr.1 . . 3 = (∥r𝑅)
43releqi 5125 . 2 (Rel ↔ Rel (∥r𝑅))
52, 4mpbir 220 1 Rel
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  .rcmulr 15769  ∥rcdsr 18461 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-dvdsr 18464 This theorem is referenced by:  dvdsr  18469  isunit  18480  subrgdvds  18617
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