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Theorem fzf 12889
Description: Establish the domain and codomain of the finite integer sequence function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
Assertion
Ref Expression
fzf ...:(ℤ × ℤ)⟶𝒫 ℤ

Proof of Theorem fzf
Dummy variables 𝑘 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zex 11978 . . . 4 ℤ ∈ V
2 ssrab2 4031 . . . 4 {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ⊆ ℤ
31, 2elpwi2 5225 . . 3 {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ
43rgen2w 3143 . 2 𝑚 ∈ ℤ ∀𝑛 ∈ ℤ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ
5 df-fz 12886 . . 3 ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)})
65fmpo 7752 . 2 (∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ ↔ ...:(ℤ × ℤ)⟶𝒫 ℤ)
74, 6mpbi 233 1 ...:(ℤ × ℤ)⟶𝒫 ℤ
Colors of variables: wff setvar class
Syntax hints:  wa 399  wcel 2114  wral 3130  {crab 3134  Vcvv 3469  𝒫 cpw 4511   class class class wbr 5042   × cxp 5530  wf 6330  cle 10665  cz 11969  ...cfz 12885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-neg 10862  df-z 11970  df-fz 12886
This theorem is referenced by:  elfz2  12892  fz0  12917  fzoval  13034  gsumval2a  17886  gsumval3  19018  topnfbey  28252
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