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Theorem plybss 23849
Description: Reverse closure of the parameter 𝑆 of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
Assertion
Ref Expression
plybss (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)

Proof of Theorem plybss
Dummy variables 𝑘 𝑎 𝑛 𝑧 𝑓 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ply 23843 . . . 4 Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))})
21dmmptss 5593 . . 3 dom Poly ⊆ 𝒫 ℂ
3 elfvdm 6178 . . 3 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ dom Poly)
42, 3sseldi 3586 . 2 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
54elpwid 4146 1 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  {cab 2612  wrex 2913  cun 3558  wss 3560  𝒫 cpw 4135  {csn 4153  cmpt 4678  dom cdm 5079  cfv 5850  (class class class)co 6605  𝑚 cmap 7803  cc 9879  0cc0 9881   · cmul 9886  0cn0 11237  ...cfz 12265  cexp 12797  Σcsu 14345  Polycply 23839
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-xp 5085  df-rel 5086  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fv 5858  df-ply 23843
This theorem is referenced by:  elply  23850  plyf  23853  plyssc  23855  plyaddlem  23870  plymullem  23871  plysub  23874  dgrlem  23884  coeidlem  23892  plyco  23896  plycj  23932  plyreres  23937  plydivlem3  23949  plydivlem4  23950  elmnc  37173
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