Theorem fncpn 25891

Description: The 𝓑C^𝑛 object is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Assertion

Ref	Expression
fncpn	⊢ (𝑆 ⊆ ℂ → (𝓑C𝑛‘𝑆) Fn ℕ0)

Proof of Theorem fncpn

Dummy variables 𝑓 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7391	. . . 4 ⊢ (ℂ ↑pm 𝑆) ∈ V
2	1	rabex 5284	. . 3 ⊢ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)} ∈ V
3		eqid 2736	. . 3 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})
4	2, 3	fnmpti 6635	. 2 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) Fn ℕ0
5		cpnfval 25890	. . 3 ⊢ (𝑆 ⊆ ℂ → (𝓑C𝑛‘𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}))
6	5	fneq1d 6585	. 2 ⊢ (𝑆 ⊆ ℂ → ((𝓑C𝑛‘𝑆) Fn ℕ0 ↔ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) Fn ℕ0))
7	4, 6	mpbiri 258	1 ⊢ (𝑆 ⊆ ℂ → (𝓑C𝑛‘𝑆) Fn ℕ0)

Syntax hints:   → wi 4   ∈ wcel 2113  {crab 3399   ⊆ wss 3901   ↦ cmpt 5179  dom cdm 5624   Fn wfn 6487  ‘cfv 6492  (class class class)co 7358   ↑_pm cpm 8764  ℂcc 11024  ℕ₀cn0 12401  –cn→ccncf 24825   D^𝑛 cdvn 25821  𝓑C^𝑛ccpn 25822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-1cn 11084  ax-addcl 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-nn 12146  df-n0 12402  df-cpn 25826

This theorem is referenced by:  cpncn  25894  cpnres  25895  plycpn  26253  aalioulem3  26298