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Mirrors > Home > ILE Home > Th. List > uzf | 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 | ssrab2 3255 | . . . 4 ⊢ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ⊆ ℤ | |
2 | zex 9291 | . . . . 5 ⊢ ℤ ∈ V | |
3 | 2 | elpw2 4175 | . . . 4 ⊢ ({𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ ↔ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ⊆ ℤ) |
4 | 1, 3 | mpbir 146 | . . 3 ⊢ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ |
5 | 4 | rgenw 2545 | . 2 ⊢ ∀𝑗 ∈ ℤ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ |
6 | df-uz 9558 | . . 3 ⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) | |
7 | 6 | fmpt 5686 | . 2 ⊢ (∀𝑗 ∈ ℤ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ ↔ ℤ≥:ℤ⟶𝒫 ℤ) |
8 | 5, 7 | mpbi 145 | 1 ⊢ ℤ≥:ℤ⟶𝒫 ℤ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 ∀wral 2468 {crab 2472 ⊆ wss 3144 𝒫 cpw 3590 class class class wbr 4018 ⟶wf 5231 ≤ cle 8022 ℤcz 9282 ℤ≥cuz 9557 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-cnex 7931 ax-resscn 7932 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-ov 5898 df-neg 8160 df-z 9283 df-uz 9558 |
This theorem is referenced by: eluzel2 9562 uzn0 9572 uzin2 11027 rexanuz 11028 climmpt 11339 lmbr2 14166 lmff 14201 |
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