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Mirrors > Home > MPE Home > Th. List > uzf | Structured version Visualization version GIF version |
Description: The domain and range of the upper integers function. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
uzf | ⊢ ℤ≥:ℤ⟶𝒫 ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3836 | . . . 4 ⊢ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ⊆ ℤ | |
2 | zex 11588 | . . . . 5 ⊢ ℤ ∈ V | |
3 | 2 | elpw2 4959 | . . . 4 ⊢ ({𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ ↔ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ⊆ ℤ) |
4 | 1, 3 | mpbir 221 | . . 3 ⊢ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ |
5 | 4 | rgenw 3073 | . 2 ⊢ ∀𝑗 ∈ ℤ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ |
6 | df-uz 11889 | . . 3 ⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) | |
7 | 6 | fmpt 6523 | . 2 ⊢ (∀𝑗 ∈ ℤ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ ↔ ℤ≥:ℤ⟶𝒫 ℤ) |
8 | 5, 7 | mpbi 220 | 1 ⊢ ℤ≥:ℤ⟶𝒫 ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2145 ∀wral 3061 {crab 3065 ⊆ wss 3723 𝒫 cpw 4297 class class class wbr 4786 ⟶wf 6027 ≤ cle 10277 ℤcz 11579 ℤ≥cuz 11888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 ax-cnex 10194 ax-resscn 10195 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-fv 6039 df-ov 6796 df-neg 10471 df-z 11580 df-uz 11889 |
This theorem is referenced by: eluzel2 11893 uzn0 11904 uzssz 11908 ltweuz 12968 uzin2 14292 rexanuz 14293 sumz 14661 sumss 14663 prod1 14881 prodss 14884 lmbr2 21284 lmff 21326 zfbas 21920 uzrest 21921 lmflf 22029 lmmbr2 23276 caucfil 23300 lmcau 23330 heibor1lem 33940 dmuz 39958 |
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