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Definition df-dm 5529
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 28224). Another example is the domain of the complex arctangent, (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i)) (for proof see atandm 25462). Contrast with range (defined in df-rn 5530). For alternate definitions see dfdm2 6100, dfdm3 5722, and dfdm4 5728. 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 5519 . 2 class dom 𝐴
3 vx . . . . . 6 setvar 𝑥
43cv 1537 . . . . 5 class 𝑥
5 vy . . . . . 6 setvar 𝑦
65cv 1537 . . . . 5 class 𝑦
74, 6, 1wbr 5030 . . . 4 wff 𝑥𝐴𝑦
87, 5wex 1781 . . 3 wff 𝑦 𝑥𝐴𝑦
98, 3cab 2776 . 2 class {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
102, 9wceq 1538 1 wff dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
Colors of variables: wff setvar class
This definition is referenced by:  dfdm3  5722  dfrn2  5723  dfdm4  5728  dfdmf  5729  eldmg  5731  dmun  5743  domepOLD  5758  dm0rn0  5759  nfdm  5787  fliftf  7047  opabdm  30375  rncossdmcoss  35855  dfatco  43810
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