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Theorem fzf 8608
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 3019 . . . 4 {𝑘 ℤ ∣ (𝑚𝑘 𝑘𝑛)} ⊆ ℤ
2 zex 7990 . . . . 5 V
32elpw2 3902 . . . 4 ({𝑘 ℤ ∣ (𝑚𝑘 𝑘𝑛)} 𝒫 ℤ ↔ {𝑘 ℤ ∣ (𝑚𝑘 𝑘𝑛)} ⊆ ℤ)
41, 3mpbir 134 . . 3 {𝑘 ℤ ∣ (𝑚𝑘 𝑘𝑛)} 𝒫 ℤ
54rgen2w 2371 . 2 𝑚 𝑛 ℤ {𝑘 ℤ ∣ (𝑚𝑘 𝑘𝑛)} 𝒫 ℤ
6 df-fz 8605 . . 3 ... = (𝑚 ℤ, 𝑛 ℤ ↦ {𝑘 ℤ ∣ (𝑚𝑘 𝑘𝑛)})
76fmpt2 5769 . 2 (𝑚 𝑛 ℤ {𝑘 ℤ ∣ (𝑚𝑘 𝑘𝑛)} 𝒫 ℤ ↔ ...:(ℤ × ℤ)⟶𝒫 ℤ)
85, 7mpbi 133 1 ...:(ℤ × ℤ)⟶𝒫 ℤ
Colors of variables: wff set class
Syntax hints:   wa 97   wcel 1390  wral 2300  {crab 2304  wss 2911  𝒫 cpw 3351   class class class wbr 3755   × cxp 4286  wf 4841  cle 6818  cz 7981  ...cfz 8604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-cnex 6734  ax-resscn 6735
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-neg 6942  df-z 7982  df-fz 8605
This theorem is referenced by:  fzen  8637  fzof  8731  fzoval  8735
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