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Theorem fzf 9428
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 3106 . . . 4 {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ⊆ ℤ
2 zex 8759 . . . . 5 ℤ ∈ V
32elpw2 3993 . . . 4 ({𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ ↔ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ⊆ ℤ)
41, 3mpbir 144 . . 3 {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ
54rgen2w 2431 . 2 𝑚 ∈ ℤ ∀𝑛 ∈ ℤ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ
6 df-fz 9425 . . 3 ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)})
76fmpt2 5971 . 2 (∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ ↔ ...:(ℤ × ℤ)⟶𝒫 ℤ)
85, 7mpbi 143 1 ...:(ℤ × ℤ)⟶𝒫 ℤ
Colors of variables: wff set class
Syntax hints:  wa 102  wcel 1438  wral 2359  {crab 2363  wss 2999  𝒫 cpw 3429   class class class wbr 3845   × cxp 4436  wf 5011  cle 7523  cz 8750  ...cfz 9424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-cnex 7436  ax-resscn 7437
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-neg 7656  df-z 8751  df-fz 9425
This theorem is referenced by:  fzen  9457  fzof  9555  fzoval  9559
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