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 5698
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 30467). Another example is the domain of the complex arctangent, (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i)) (for proof see atandm 26933). Contrast with range (defined in df-rn 5699). For alternate definitions see dfdm2 6302, dfdm3 5900, and dfdm4 5908. 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 5688 . 2 class dom 𝐴
3 vx . . . . . 6 setvar 𝑥
43cv 1535 . . . . 5 class 𝑥
5 vy . . . . . 6 setvar 𝑦
65cv 1535 . . . . 5 class 𝑦
74, 6, 1wbr 5147 . . . 4 wff 𝑥𝐴𝑦
87, 5wex 1775 . . 3 wff 𝑦 𝑥𝐴𝑦
98, 3cab 2711 . 2 class {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
102, 9wceq 1536 1 wff dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
Colors of variables: wff setvar class
This definition is referenced by:  dfdm3  5900  dfrn2  5901  dfdm4  5908  dfdmf  5909  eldmg  5911  dmun  5923  dm0rn0  5937  nfdm  5964  fliftf  7334  opabdm  32630  rncossdmcoss  38436  dfatco  47205
  Copyright terms: Public domain W3C validator