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Theorem fzf 12537
 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 ssrab2 3836 . . . 4 {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ⊆ ℤ
2 zex 11588 . . . . 5 ℤ ∈ V
32elpw2 4959 . . . 4 ({𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ ↔ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ⊆ ℤ)
41, 3mpbir 221 . . 3 {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ
54rgen2w 3074 . 2 𝑚 ∈ ℤ ∀𝑛 ∈ ℤ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ
6 df-fz 12534 . . 3 ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)})
76fmpt2 7387 . 2 (∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ ↔ ...:(ℤ × ℤ)⟶𝒫 ℤ)
85, 7mpbi 220 1 ...:(ℤ × ℤ)⟶𝒫 ℤ
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 382   ∈ wcel 2145  ∀wral 3061  {crab 3065   ⊆ wss 3723  𝒫 cpw 4297   class class class wbr 4786   × cxp 5247  ⟶wf 6027   ≤ cle 10277  ℤcz 11579  ...cfz 12533 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-neg 10471  df-z 11580  df-fz 12534 This theorem is referenced by:  elfz2  12540  fz0  12563  fzoval  12679  gsumval2a  17487  gsumval3  18515  topnfbey  27667
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