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Theorem plybss 23754
 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 23748 . . . 4 Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0𝑎 ∈ ((𝑥 ∪ {0}) ↑𝑚0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎𝑘) · (𝑧𝑘)))})
21dmmptss 5548 . . 3 dom Poly ⊆ 𝒫 ℂ
3 elfvdm 6130 . . 3 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ dom Poly)
42, 3sseldi 3566 . 2 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
54elpwid 4118 1 (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897   ∪ cun 3538   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125   ↦ cmpt 4643  dom cdm 5038  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  ℂcc 9813  0cc0 9815   · cmul 9820  ℕ0cn0 11169  ...cfz 12197  ↑cexp 12722  Σcsu 14264  Polycply 23744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fv 5812  df-ply 23748 This theorem is referenced by:  elply  23755  plyf  23758  plyssc  23760  plyaddlem  23775  plymullem  23776  plysub  23779  dgrlem  23789  coeidlem  23797  plyco  23801  plycj  23837  plyreres  23842  plydivlem3  23854  plydivlem4  23855  elmnc  36725
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