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Mirrors > Home > MPE Home > Th. List > uzval | Structured version Visualization version GIF version |
Description: The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
uzval | ⊢ (𝑁 ∈ ℤ → (ℤ≥‘𝑁) = {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5033 | . . 3 ⊢ (𝑗 = 𝑁 → (𝑗 ≤ 𝑘 ↔ 𝑁 ≤ 𝑘)) | |
2 | 1 | rabbidv 3427 | . 2 ⊢ (𝑗 = 𝑁 → {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} = {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘}) |
3 | df-uz 12232 | . 2 ⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) | |
4 | zex 11978 | . . 3 ⊢ ℤ ∈ V | |
5 | 4 | rabex 5199 | . 2 ⊢ {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} ∈ V |
6 | 2, 3, 5 | fvmpt 6745 | 1 ⊢ (𝑁 ∈ ℤ → (ℤ≥‘𝑁) = {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {crab 3110 class class class wbr 5030 ‘cfv 6324 ≤ cle 10665 ℤcz 11969 ℤ≥cuz 12231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-neg 10862 df-z 11970 df-uz 12232 |
This theorem is referenced by: eluz1 12235 nn0uz 12268 nnuz 12269 algfx 15914 |
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