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Theorem reldvdsr 19662
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 19659 . . 3 r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r𝑤)𝑥) = 𝑦)})
21relmptopab 7455 . 2 Rel (∥r𝑅)
3 reldvdsr.1 . . 3 = (∥r𝑅)
43releqi 5649 . 2 (Rel ↔ Rel (∥r𝑅))
52, 4mpbir 234 1 Rel
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1543  wcel 2110  wrex 3062  Vcvv 3408  Rel wrel 5556  cfv 6380  (class class class)co 7213  Basecbs 16760  .rcmulr 16803  rcdsr 19656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fv 6388  df-dvdsr 19659
This theorem is referenced by:  dvdsr  19664  isunit  19675  subrgdvds  19814
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