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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 9791, which includes 𝑁. Not including the endpoint simplifies a number of formulas related to cardinality and splitting; contrast fzosplit 9954 with fzsplit 9831, 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 9919 | . 2 class ..^ | |
2 | vm | . . 3 setvar 𝑚 | |
3 | vn | . . 3 setvar 𝑛 | |
4 | cz 9054 | . . 3 class ℤ | |
5 | 2 | cv 1330 | . . . 4 class 𝑚 |
6 | 3 | cv 1330 | . . . . 5 class 𝑛 |
7 | c1 7621 | . . . . 5 class 1 | |
8 | cmin 7933 | . . . . 5 class − | |
9 | 6, 7, 8 | co 5774 | . . . 4 class (𝑛 − 1) |
10 | cfz 9790 | . . . 4 class ... | |
11 | 5, 9, 10 | co 5774 | . . 3 class (𝑚...(𝑛 − 1)) |
12 | 2, 3, 4, 4, 11 | cmpo 5776 | . 2 class (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) |
13 | 1, 12 | wceq 1331 | 1 wff ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) |
Colors of variables: wff set class |
This definition is referenced by: fzof 9921 elfzoel1 9922 elfzoel2 9923 fzoval 9925 |
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