Theorem plybss 26099

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 26093	. . 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 6977	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
3	2	elpwid 4572	1 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053   ∪ cun 3912   ⊆ wss 3914  𝒫 cpw 4563  {csn 4589   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387   ↑_m cmap 8799  ℂcc 11066  0cc0 11068   · cmul 11073  ℕ₀cn0 12442  ...cfz 13468  ↑cexp 14026  Σcsu 15652  Polycply 26089

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 5251  ax-nul 5261  ax-pr 5387

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 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fv 6519  df-ply 26093

This theorem is referenced by:  elply  26100  plyf  26103  plyssc  26105  plyaddlem  26120  plymullem  26121  plysub  26124  dgrlem  26134  coeidlem  26142  plyco  26146  plycj  26183  plycjOLD  26185  plyreres  26190  plydivlem3  26203  plydivlem4  26204  elmnc  43125