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.

Step	Hyp	Ref	Expression
1		df-ply 24285	. . . 4 ⊢ Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
2	1	dmmptss 5850	. . 3 ⊢ dom Poly ⊆ 𝒫 ℂ
3		elfvdm 6443	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ dom Poly)
4	2, 3	sseldi 3796	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
5	4	elpwid 4361	1 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)

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  ℂcc 10222  0cc0 10224   · cmul 10229  ℕ₀cn0 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