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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 7048 | . . . 4 ⊢ (ℂ ↑pm 𝑆) ∈ V | |
2 | 1 | rabex 5126 | . . 3 ⊢ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)} ∈ V |
3 | eqid 2795 | . . 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 6359 | . 2 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)}) Fn ℕ0 |
5 | cpnfval 24212 | . . 3 ⊢ (𝑆 ⊆ ℂ → (Cn‘𝑆) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (ℂ ↑pm 𝑆) ∣ ((𝑆 D𝑛 𝑓)‘𝑛) ∈ (dom 𝑓–cn→ℂ)})) | |
6 | 5 | fneq1d 6316 | . 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 259 | 1 ⊢ (𝑆 ⊆ ℂ → (Cn‘𝑆) Fn ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2081 {crab 3109 ⊆ wss 3859 ↦ cmpt 5041 dom cdm 5443 Fn wfn 6220 ‘cfv 6225 (class class class)co 7016 ↑pm cpm 8257 ℂcc 10381 ℕ0cn0 11745 –cn→ccncf 23167 D𝑛 cdvn 24145 Cnccpn 24146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-1cn 10441 ax-addcl 10443 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ov 7019 df-om 7437 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-nn 11487 df-n0 11746 df-cpn 24150 |
This theorem is referenced by: cpncn 24216 cpnres 24217 plycpn 24561 aalioulem3 24606 |
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