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Mirrors > Home > ILE Home > Th. List > uzf | 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 3222 | . . . 4 ⊢ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ⊆ ℤ | |
2 | zex 9191 | . . . . 5 ⊢ ℤ ∈ V | |
3 | 2 | elpw2 4130 | . . . 4 ⊢ ({𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ ↔ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ⊆ ℤ) |
4 | 1, 3 | mpbir 145 | . . 3 ⊢ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ |
5 | 4 | rgenw 2519 | . 2 ⊢ ∀𝑗 ∈ ℤ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ |
6 | df-uz 9458 | . . 3 ⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) | |
7 | 6 | fmpt 5629 | . 2 ⊢ (∀𝑗 ∈ ℤ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} ∈ 𝒫 ℤ ↔ ℤ≥:ℤ⟶𝒫 ℤ) |
8 | 5, 7 | mpbi 144 | 1 ⊢ ℤ≥:ℤ⟶𝒫 ℤ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2135 ∀wral 2442 {crab 2446 ⊆ wss 3111 𝒫 cpw 3553 class class class wbr 3976 ⟶wf 5178 ≤ cle 7925 ℤcz 9182 ℤ≥cuz 9457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-cnex 7835 ax-resscn 7836 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-ov 5839 df-neg 8063 df-z 9183 df-uz 9458 |
This theorem is referenced by: eluzel2 9462 uzn0 9472 uzin2 10915 rexanuz 10916 climmpt 11227 lmbr2 12761 lmff 12796 |
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