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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 26688). Another example is the domain of the complex arctangent, (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i)) (for proof see atandm 24403). Contrast with range (defined in df-rn 5049). For alternate definitions see dfdm2 5584, dfdm3 5232, and dfdm4 5238. 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 5038 | . 2 class dom 𝐴 |
3 | vx | . . . . . 6 setvar 𝑥 | |
4 | 3 | cv 1474 | . . . . 5 class 𝑥 |
5 | vy | . . . . . 6 setvar 𝑦 | |
6 | 5 | cv 1474 | . . . . 5 class 𝑦 |
7 | 4, 6, 1 | wbr 4583 | . . . 4 wff 𝑥𝐴𝑦 |
8 | 7, 5 | wex 1695 | . . 3 wff ∃𝑦 𝑥𝐴𝑦 |
9 | 8, 3 | cab 2596 | . 2 class {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
10 | 2, 9 | wceq 1475 | 1 wff dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
Colors of variables: wff setvar class |
This definition is referenced by: dfdm3 5232 dfrn2 5233 dfdm4 5238 dfdmf 5239 eldmg 5241 dmun 5253 dm0rn0 5263 nfdm 5288 fliftf 6465 opabdm 28803 domep 30942 |
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