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Theorem mplrcl 19430
Description: Reverse closure for the polynomial index set. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
Hypotheses
Ref Expression
mplrcl.p 𝑃 = (𝐼 mPoly 𝑅)
mplrcl.b 𝐵 = (Base‘𝑃)
Assertion
Ref Expression
mplrcl (𝑋𝐵𝐼 ∈ V)

Proof of Theorem mplrcl
StepHypRef Expression
1 mplrcl.p . 2 𝑃 = (𝐼 mPoly 𝑅)
2 mplrcl.b . 2 𝐵 = (Base‘𝑃)
3 reldmmpl 19367 . 2 Rel dom mPoly
41, 2, 3strov2rcl 15862 1 (𝑋𝐵𝐼 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3190  cfv 5857  (class class class)co 6615  Basecbs 15800   mPoly cmpl 19293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-slot 15804  df-base 15805  df-mpl 19298
This theorem is referenced by:  mdegleb  23762  mdeglt  23763  mdegldg  23764  mdegxrcl  23765  mdegcl  23767  mdegnn0cl  23769
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