MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-dm Structured version   Visualization version   GIF version

Definition df-dm 5321
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 27623). Another example is the domain of the complex arctangent, (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i)) (for proof see atandm 24813). Contrast with range (defined in df-rn 5322). For alternate definitions see dfdm2 5881, dfdm3 5511, and dfdm4 5517. The notation "dom " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
df-dm dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
Distinct variable group:   𝑥,𝑦,𝐴

Detailed syntax breakdown of Definition df-dm
StepHypRef Expression
1 cA . . 3 class 𝐴
21cdm 5311 . 2 class dom 𝐴
3 vx . . . . . 6 setvar 𝑥
43cv 1636 . . . . 5 class 𝑥
5 vy . . . . . 6 setvar 𝑦
65cv 1636 . . . . 5 class 𝑦
74, 6, 1wbr 4844 . . . 4 wff 𝑥𝐴𝑦
87, 5wex 1859 . . 3 wff 𝑦 𝑥𝐴𝑦
98, 3cab 2792 . 2 class {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
102, 9wceq 1637 1 wff dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
Colors of variables: wff setvar class
This definition is referenced by:  dfdm3  5511  dfrn2  5512  dfdm4  5517  dfdmf  5518  eldmg  5520  dmun  5532  dm0rn0  5543  nfdm  5568  fliftf  6785  opabdm  29744  domep  32013  rncossdmcoss  34513  dfatco  41839
  Copyright terms: Public domain W3C validator