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Theorem mplrcl 19811
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 19749 . 2 Rel dom mPoly
41, 2, 3strov2rcl 16246 1 (𝑋𝐵𝐼 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  Vcvv 3386  cfv 6102  (class class class)co 6879  Basecbs 16183   mPoly cmpl 19675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-iota 6065  df-fun 6104  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-slot 16187  df-base 16189  df-mpl 19680
This theorem is referenced by:  mdegleb  24164  mdeglt  24165  mdegldg  24166  mdegxrcl  24167  mdegcl  24169  mdegnn0cl  24171
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