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Mirrors > Home > ILE Home > Th. List > fzval | GIF version |
Description: The value of a finite set of sequential integers. E.g., 2...5 means the set {2, 3, 4, 5}. A special case of this definition (starting at 1) appears as Definition 11-2.1 of [Gleason] p. 141, where ℕk means our 1...𝑘; he calls these sets segments of the integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
fzval | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 3932 | . . . 4 ⊢ (𝑚 = 𝑀 → (𝑚 ≤ 𝑘 ↔ 𝑀 ≤ 𝑘)) | |
2 | 1 | anbi1d 460 | . . 3 ⊢ (𝑚 = 𝑀 → ((𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ↔ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛))) |
3 | 2 | rabbidv 2675 | . 2 ⊢ (𝑚 = 𝑀 → {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)}) |
4 | breq2 3933 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑘 ≤ 𝑛 ↔ 𝑘 ≤ 𝑁)) | |
5 | 4 | anbi2d 459 | . . 3 ⊢ (𝑛 = 𝑁 → ((𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛) ↔ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁))) |
6 | 5 | rabbidv 2675 | . 2 ⊢ (𝑛 = 𝑁 → {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) |
7 | df-fz 9791 | . 2 ⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)}) | |
8 | zex 9063 | . . 3 ⊢ ℤ ∈ V | |
9 | 8 | rabex 4072 | . 2 ⊢ {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)} ∈ V |
10 | 3, 6, 7, 9 | ovmpo 5906 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 {crab 2420 class class class wbr 3929 (class class class)co 5774 ≤ cle 7801 ℤcz 9054 ...cfz 9790 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-neg 7936 df-z 9055 df-fz 9791 |
This theorem is referenced by: fzval2 9793 elfz1 9795 fznlem 9821 |
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