Theorem fzf 10394

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.

Step	Hyp	Ref	Expression
1		ssrab2 3333	. . . 4 ⊢ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} ⊆ ℤ
2		zex 9632	. . . . 5 ⊢ ℤ ∈ V
3	2	elpw2 4288	. . . 4 ⊢ ({𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} ∈ 𝒫 ℤ ↔ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} ⊆ ℤ)
4	1, 3	mpbir 146	. . 3 ⊢ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} ∈ 𝒫 ℤ
5	4	rgen2w 2606	. 2 ⊢ ∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} ∈ 𝒫 ℤ
6		df-fz 10391	. . 3 ⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)})
7	6	fmpo 6427	. 2 ⊢ (∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} ∈ 𝒫 ℤ ↔ ...:(ℤ × ℤ)⟶𝒫 ℤ)
8	5, 7	mpbi 145	1 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ

Syntax hints:   ∧ wa 104   ∈ wcel 2209  ∀wral 2528  {crab 2532   ⊆ wss 3220  𝒫 cpw 3685   class class class wbr 4125   × cxp 4767  ⟶wf 5368   ≤ cle 8351  ℤcz 9623  ...cfz 10390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-neg 8490  df-z 9624  df-fz 10391

This theorem is referenced by:  fzen  10426  fzof  10529  fzoval  10533