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Mirrors > Home > MPE Home > Th. List > fzf | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
fzf | ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zex 12466 | . . . 4 ⊢ ℤ ∈ V | |
2 | ssrab2 4035 | . . . 4 ⊢ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} ⊆ ℤ | |
3 | 1, 2 | elpwi2 5301 | . . 3 ⊢ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} ∈ 𝒫 ℤ |
4 | 3 | rgen2w 3067 | . 2 ⊢ ∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} ∈ 𝒫 ℤ |
5 | df-fz 13379 | . . 3 ⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)}) | |
6 | 5 | fmpo 7992 | . 2 ⊢ (∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} ∈ 𝒫 ℤ ↔ ...:(ℤ × ℤ)⟶𝒫 ℤ) |
7 | 4, 6 | mpbi 229 | 1 ⊢ ...:(ℤ × ℤ)⟶𝒫 ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∈ wcel 2106 ∀wral 3062 {crab 3405 Vcvv 3443 𝒫 cpw 4558 class class class wbr 5103 × cxp 5629 ⟶wf 6489 ≤ cle 11148 ℤcz 12457 ...cfz 13378 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-neg 11346 df-z 12458 df-fz 13379 |
This theorem is referenced by: elfz2 13385 fz0 13410 fzoval 13527 gsumval2a 18499 gsumval3 19642 topnfbey 29241 |
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