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Theorem fzf 13548
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 12620 . . . 4 ℤ ∈ V
2 ssrab2 4090 . . . 4 {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ⊆ ℤ
31, 2elpwi2 5341 . . 3 {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ
43rgen2w 3064 . 2 𝑚 ∈ ℤ ∀𝑛 ∈ ℤ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ
5 df-fz 13545 . . 3 ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)})
65fmpo 8092 . 2 (∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ ↔ ...:(ℤ × ℤ)⟶𝒫 ℤ)
74, 6mpbi 230 1 ...:(ℤ × ℤ)⟶𝒫 ℤ
Colors of variables: wff setvar class
Syntax hints:  wa 395  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  𝒫 cpw 4605   class class class wbr 5148   × cxp 5687  wf 6559  cle 11294  cz 12611  ...cfz 13544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-neg 11493  df-z 12612  df-fz 13545
This theorem is referenced by:  elfz2  13551  fz0  13576  fzoval  13697  gsumval2a  18711  gsumval3  19940  topnfbey  30498
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