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| Mirrors > Home > MPE Home > Th. List > df-dm | Structured version Visualization version GIF version | ||
| Description: Define the domain of a class. Definition 3 of [Suppes] p. 59. For example, 𝐹 = {〈2, 6〉, 〈3, 9〉} → dom 𝐹 = {2, 3} (ex-dm 30458). Another example is the domain of the complex arctangent, (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i)) (for proof see atandm 26919). Contrast with range (defined in df-rn 5696). For alternate definitions see dfdm2 6301, dfdm3 5898, and dfdm4 5906. The notation "dom " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.) |
| Ref | Expression |
|---|---|
| df-dm | ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cA | . . 3 class 𝐴 | |
| 2 | 1 | cdm 5685 | . 2 class dom 𝐴 |
| 3 | vx | . . . . . 6 setvar 𝑥 | |
| 4 | 3 | cv 1539 | . . . . 5 class 𝑥 |
| 5 | vy | . . . . . 6 setvar 𝑦 | |
| 6 | 5 | cv 1539 | . . . . 5 class 𝑦 |
| 7 | 4, 6, 1 | wbr 5143 | . . . 4 wff 𝑥𝐴𝑦 |
| 8 | 7, 5 | wex 1779 | . . 3 wff ∃𝑦 𝑥𝐴𝑦 |
| 9 | 8, 3 | cab 2714 | . 2 class {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
| 10 | 2, 9 | wceq 1540 | 1 wff dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
| Colors of variables: wff setvar class |
| This definition is referenced by: dfdm3 5898 dfrn2 5899 dfdm4 5906 dfdmf 5907 eldmg 5909 dmun 5921 dm0rn0 5935 nfdm 5962 fliftf 7335 opabdm 32623 rncossdmcoss 38456 dfatco 47268 |
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