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Mirrors > Home > ILE Home > Th. List > uzval | 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 3969 | . . 3 ⊢ (𝑗 = 𝑁 → (𝑗 ≤ 𝑘 ↔ 𝑁 ≤ 𝑘)) | |
2 | 1 | rabbidv 2701 | . 2 ⊢ (𝑗 = 𝑁 → {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} = {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘}) |
3 | df-uz 9441 | . 2 ⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) | |
4 | zex 9177 | . . 3 ⊢ ℤ ∈ V | |
5 | 4 | rabex 4109 | . 2 ⊢ {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} ∈ V |
6 | 2, 3, 5 | fvmpt 5546 | 1 ⊢ (𝑁 ∈ ℤ → (ℤ≥‘𝑁) = {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 ∈ wcel 2128 {crab 2439 class class class wbr 3966 ‘cfv 5171 ≤ cle 7914 ℤcz 9168 ℤ≥cuz 9440 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-cnex 7824 ax-resscn 7825 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-iota 5136 df-fun 5173 df-fv 5179 df-ov 5828 df-neg 8050 df-z 9169 df-uz 9441 |
This theorem is referenced by: eluz1 9444 nn0uz 9474 nnuz 9475 algfx 11933 |
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