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Mirrors > Home > ILE Home > Th. List > df-fzo | GIF version |
Description: Define a function generating sets of integers using a half-open range. Read (𝑀..^𝑁) as the integers from 𝑀 up to, but not including, 𝑁; contrast with (𝑀...𝑁) df-fz 9966, which includes 𝑁. Not including the endpoint simplifies a number of formulas related to cardinality and splitting; contrast fzosplit 10133 with fzsplit 10007, for instance. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
Ref | Expression |
---|---|
df-fzo | ⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cfzo 10098 | . 2 class ..^ | |
2 | vm | . . 3 setvar 𝑚 | |
3 | vn | . . 3 setvar 𝑛 | |
4 | cz 9212 | . . 3 class ℤ | |
5 | 2 | cv 1347 | . . . 4 class 𝑚 |
6 | 3 | cv 1347 | . . . . 5 class 𝑛 |
7 | c1 7775 | . . . . 5 class 1 | |
8 | cmin 8090 | . . . . 5 class − | |
9 | 6, 7, 8 | co 5853 | . . . 4 class (𝑛 − 1) |
10 | cfz 9965 | . . . 4 class ... | |
11 | 5, 9, 10 | co 5853 | . . 3 class (𝑚...(𝑛 − 1)) |
12 | 2, 3, 4, 4, 11 | cmpo 5855 | . 2 class (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) |
13 | 1, 12 | wceq 1348 | 1 wff ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) |
Colors of variables: wff set class |
This definition is referenced by: fzof 10100 elfzoel1 10101 elfzoel2 10102 fzoval 10104 |
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