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Theorem reldvdsr 18997
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 18994 . . 3 r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r𝑤)𝑥) = 𝑦)})
21relmptopab 7142 . 2 Rel (∥r𝑅)
3 reldvdsr.1 . . 3 = (∥r𝑅)
43releqi 5436 . 2 (Rel ↔ Rel (∥r𝑅))
52, 4mpbir 223 1 Rel
Colors of variables: wff setvar class
Syntax hints:  wa 386   = wceq 1658  wcel 2166  wrex 3117  Vcvv 3413  Rel wrel 5346  cfv 6122  (class class class)co 6904  Basecbs 16221  .rcmulr 16305  rcdsr 18991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fv 6130  df-dvdsr 18994
This theorem is referenced by:  dvdsr  18999  isunit  19010  subrgdvds  19149
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