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Mirrors > Home > ILE Home > Th. List > fzf | 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 | ssrab2 3241 | . . . 4 ⊢ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} ⊆ ℤ | |
2 | zex 9262 | . . . . 5 ⊢ ℤ ∈ V | |
3 | 2 | elpw2 4158 | . . . 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 2533 | . 2 ⊢ ∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} ∈ 𝒫 ℤ |
6 | df-fz 10009 | . . 3 ⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)}) | |
7 | 6 | fmpo 6202 | . 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 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∈ wcel 2148 ∀wral 2455 {crab 2459 ⊆ wss 3130 𝒫 cpw 3576 class class class wbr 4004 × cxp 4625 ⟶wf 5213 ≤ cle 7993 ℤcz 9253 ...cfz 10008 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-cnex 7902 ax-resscn 7903 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-fv 5225 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-neg 8131 df-z 9254 df-fz 10009 |
This theorem is referenced by: fzen 10043 fzof 10144 fzoval 10148 |
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