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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 | zex 12328 | . . . 4 ⊢ ℤ ∈ V | |
2 | ssrab2 4013 | . . . 4 ⊢ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ⊆ ℤ | |
3 | 1, 2 | elpwi2 5270 | . . 3 ⊢ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ |
4 | 3 | rgenw 3076 | . 2 ⊢ ∀𝑗 ∈ ℤ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ |
5 | df-uz 12583 | . . 3 ⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) | |
6 | 5 | fmpt 6984 | . 2 ⊢ (∀𝑗 ∈ ℤ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ ↔ ℤ≥:ℤ⟶𝒫 ℤ) |
7 | 4, 6 | mpbi 229 | 1 ⊢ ℤ≥:ℤ⟶𝒫 ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ∀wral 3064 {crab 3068 Vcvv 3432 𝒫 cpw 4533 class class class wbr 5074 ⟶wf 6429 ≤ cle 11010 ℤcz 12319 ℤ≥cuz 12582 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-cnex 10927 ax-resscn 10928 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-neg 11208 df-z 12320 df-uz 12583 |
This theorem is referenced by: eluzel2 12587 uzn0 12599 uzssz 12603 ltweuz 13681 uzin2 15056 rexanuz 15057 sumz 15434 sumss 15436 prod1 15654 prodss 15657 lmbr2 22410 lmff 22452 zfbas 23047 uzrest 23048 lmflf 23156 lmmbr2 24423 caucfil 24447 lmcau 24477 heibor1lem 35967 dmuz 42777 |
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