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Theorem dvdsrcl 20311
Description: Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
Hypotheses
Ref Expression
dvdsr.1 𝐵 = (Base‘𝑅)
dvdsr.2 = (∥r𝑅)
Assertion
Ref Expression
dvdsrcl (𝑋 𝑌𝑋𝐵)

Proof of Theorem dvdsrcl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dvdsr.1 . . 3 𝐵 = (Base‘𝑅)
2 dvdsr.2 . . 3 = (∥r𝑅)
3 eqid 2728 . . 3 (.r𝑅) = (.r𝑅)
41, 2, 3dvdsr 20308 . 2 (𝑋 𝑌 ↔ (𝑋𝐵 ∧ ∃𝑥𝐵 (𝑥(.r𝑅)𝑋) = 𝑌))
54simplbi 496 1 (𝑋 𝑌𝑋𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wrex 3067   class class class wbr 5152  cfv 6553  (class class class)co 7426  Basecbs 17187  .rcmulr 17241  rcdsr 20300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-dvdsr 20303
This theorem is referenced by:  unitcl  20321  rprmasso  33267
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