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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 3992 | . . 3 ⊢ (𝑗 = 𝑁 → (𝑗 ≤ 𝑘 ↔ 𝑁 ≤ 𝑘)) | |
2 | 1 | rabbidv 2719 | . 2 ⊢ (𝑗 = 𝑁 → {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} = {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘}) |
3 | df-uz 9488 | . 2 ⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) | |
4 | zex 9221 | . . 3 ⊢ ℤ ∈ V | |
5 | 4 | rabex 4133 | . 2 ⊢ {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} ∈ V |
6 | 2, 3, 5 | fvmpt 5573 | 1 ⊢ (𝑁 ∈ ℤ → (ℤ≥‘𝑁) = {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 {crab 2452 class class class wbr 3989 ‘cfv 5198 ≤ cle 7955 ℤcz 9212 ℤ≥cuz 9487 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-cnex 7865 ax-resscn 7866 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-ov 5856 df-neg 8093 df-z 9213 df-uz 9488 |
This theorem is referenced by: eluz1 9491 nn0uz 9521 nnuz 9522 algfx 12006 |
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