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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 3847 . . . 4 {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ⊆ ℤ
2 zex 11633 . . . . 5 ℤ ∈ V
32elpw2 4986 . . . 4 ({𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ ↔ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ⊆ ℤ)
41, 3mpbir 222 . . 3 {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ
54rgen2w 3072 . 2 𝑚 ∈ ℤ ∀𝑛 ∈ ℤ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ
6 df-fz 12534 . . 3 ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)})
76fmpt2 7438 . 2 (∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} ∈ 𝒫 ℤ ↔ ...:(ℤ × ℤ)⟶𝒫 ℤ)
85, 7mpbi 221 1 ...:(ℤ × ℤ)⟶𝒫 ℤ
Colors of variables: wff setvar class
Syntax hints:  wa 384  wcel 2155  wral 3055  {crab 3059  wss 3732  𝒫 cpw 4315   class class class wbr 4809   × cxp 5275  wf 6064  cle 10329  cz 11624  ...cfz 12533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-neg 10523  df-z 11625  df-fz 12534
This theorem is referenced by:  elfz2  12540  fz0  12563  fzoval  12679  gsumval2a  17547  gsumval3  18574  topnfbey  27719
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