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Mirrors > Home > MPE Home > Th. List > fncpn | Structured version Visualization version GIF version |
Description: The Cn object is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
fncpn | ⊢ (𝑆 ⊆ ℂ → (Cn‘𝑆) Fn ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 6910 | . . . 4 ⊢ (ℂ ↑pm 𝑆) ∈ V | |
2 | 1 | rabex 5007 | . . 3 ⊢ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)} ∈ V |
3 | eqid 2799 | . . 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 6233 | . 2 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) Fn ℕ0 |
5 | cpnfval 24036 | . . 3 ⊢ (𝑆 ⊆ ℂ → (Cn‘𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})) | |
6 | 5 | fneq1d 6192 | . 2 ⊢ (𝑆 ⊆ ℂ → ((Cn‘𝑆) Fn ℕ0 ↔ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) Fn ℕ0)) |
7 | 4, 6 | mpbiri 250 | 1 ⊢ (𝑆 ⊆ ℂ → (Cn‘𝑆) Fn ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2157 {crab 3093 ⊆ wss 3769 ↦ cmpt 4922 dom cdm 5312 Fn wfn 6096 ‘cfv 6101 (class class class)co 6878 ↑pm cpm 8096 ℂcc 10222 ℕ0cn0 11580 –cn→ccncf 23007 D𝑛 cdvn 23969 Cnccpn 23970 |
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-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-1cn 10282 ax-addcl 10284 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 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-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-nn 11313 df-n0 11581 df-cpn 23974 |
This theorem is referenced by: cpncn 24040 cpnres 24041 plycpn 24385 aalioulem3 24430 |
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