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| Mirrors > Home > MPE Home > Th. List > uzf | Structured version Visualization version GIF version | ||
| Description: The domain and codomain 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 | zex 12624 | . . . 4 ⊢ ℤ ∈ V | |
| 2 | ssrab2 4079 | . . . 4 ⊢ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ⊆ ℤ | |
| 3 | 1, 2 | elpwi2 5334 | . . 3 ⊢ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ | 
| 4 | 3 | rgenw 3064 | . 2 ⊢ ∀𝑗 ∈ ℤ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ | 
| 5 | df-uz 12880 | . . 3 ⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) | |
| 6 | 5 | fmpt 7129 | . 2 ⊢ (∀𝑗 ∈ ℤ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ ↔ ℤ≥:ℤ⟶𝒫 ℤ) | 
| 7 | 4, 6 | mpbi 230 | 1 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2107 ∀wral 3060 {crab 3435 Vcvv 3479 𝒫 cpw 4599 class class class wbr 5142 ⟶wf 6556 ≤ cle 11297 ℤcz 12615 ℤ≥cuz 12879 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-cnex 11212 ax-resscn 11213 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-neg 11496 df-z 12616 df-uz 12880 | 
| This theorem is referenced by: eluzel2 12884 uzn0 12896 uzssz 12900 ltweuz 14003 uzin2 15384 rexanuz 15385 sumz 15759 sumss 15761 prod1 15981 prodss 15984 lmbr2 23268 lmff 23310 zfbas 23905 uzrest 23906 lmflf 24014 lmmbr2 25294 caucfil 25318 lmcau 25348 heibor1lem 37817 dmuz 45244 | 
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