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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 11984 | . . . 4 ⊢ ℤ ∈ V | |
2 | ssrab2 4056 | . . . 4 ⊢ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ⊆ ℤ | |
3 | 1, 2 | elpwi2 5242 | . . 3 ⊢ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ |
4 | 3 | rgenw 3150 | . 2 ⊢ ∀𝑗 ∈ ℤ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ |
5 | df-uz 12238 | . . 3 ⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) | |
6 | 5 | fmpt 6869 | . 2 ⊢ (∀𝑗 ∈ ℤ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ ↔ ℤ≥:ℤ⟶𝒫 ℤ) |
7 | 4, 6 | mpbi 232 | 1 ⊢ ℤ≥:ℤ⟶𝒫 ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 ∀wral 3138 {crab 3142 Vcvv 3495 𝒫 cpw 4539 class class class wbr 5059 ⟶wf 6346 ≤ cle 10670 ℤcz 11975 ℤ≥cuz 12237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-cnex 10587 ax-resscn 10588 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-ov 7153 df-neg 10867 df-z 11976 df-uz 12238 |
This theorem is referenced by: eluzel2 12242 uzn0 12254 uzssz 12258 ltweuz 13323 uzin2 14698 rexanuz 14699 sumz 15073 sumss 15075 prod1 15292 prodss 15295 lmbr2 21861 lmff 21903 zfbas 22498 uzrest 22499 lmflf 22607 lmmbr2 23856 caucfil 23880 lmcau 23910 heibor1lem 35081 dmuz 41496 |
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