Theorem plybss 26132

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 26126	. . 3 ⊢ Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0 ∃𝑎 ∈ ((𝑥 ∪ {0}) ↑m ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))})
2	1	mptrcl 6959	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
3	2	elpwid 4568	1 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053   ∪ cun 3909   ⊆ wss 3911  𝒫 cpw 4559  {csn 4585   ↦ cmpt 5183  ‘cfv 6499  (class class class)co 7369   ↑_m cmap 8776  ℂcc 11042  0cc0 11044   · cmul 11049  ℕ₀cn0 12418  ...cfz 13444  ↑cexp 14002  Σcsu 15628  Polycply 26122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fv 6507  df-ply 26126

This theorem is referenced by:  elply  26133  plyf  26136  plyssc  26138  plyaddlem  26153  plymullem  26154  plysub  26157  dgrlem  26167  coeidlem  26175  plyco  26179  plycj  26216  plycjOLD  26218  plyreres  26223  plydivlem3  26236  plydivlem4  26237  elmnc  43118