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Theorem plybss 24291
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 24285 . . . 4 Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))})
21dmmptss 5850 . . 3 dom Poly ⊆ 𝒫 ℂ
3 elfvdm 6443 . . 3 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ dom Poly)
42, 3sseldi 3796 . 2 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
54elpwid 4361 1 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  {cab 2785  wrex 3090  cun 3767  wss 3769  𝒫 cpw 4349  {csn 4368  cmpt 4922  dom cdm 5312  cfv 6101  (class class class)co 6878  𝑚 cmap 8095  cc 10222  0cc0 10224   · cmul 10229  0cn0 11580  ...cfz 12580  cexp 13114  Σcsu 14757  Polycply 24281
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 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
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 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-xp 5318  df-rel 5319  df-cnv 5320  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fv 6109  df-ply 24285
This theorem is referenced by:  elply  24292  plyf  24295  plyssc  24297  plyaddlem  24312  plymullem  24313  plysub  24316  dgrlem  24326  coeidlem  24334  plyco  24338  plycj  24374  plyreres  24379  plydivlem3  24391  plydivlem4  24392  elmnc  38491
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