Definition df-dm 5665

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 31067). Another example is the domain of the complex arctangent, (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i)) (for proof see atandm 27231). Contrast with range (defined in df-rn 5666). For alternate definitions see dfdm2 6299, dfdm3 5875, and dfdm4 5883. 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

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2	1	cdm 5655	. 2 class dom 𝐴
3		vx	. . . . . 6 setvar 𝑥
4	3	cv 1569	. . . . 5 class 𝑥
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1569	. . . . 5 class 𝑦
7	4, 6, 1	wbr 5106	. . . 4 wff 𝑥𝐴𝑦
8	7, 5	wex 1812	. . 3 wff ∃𝑦 𝑥𝐴𝑦
9	8, 3	cab 2741	. 2 class {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
10	2, 9	wceq 1570	1 wff dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}

This definition is used by:  dfdm3  5875  dfrn2  5876  dfdm4  5883  dfdmf  5884  eldmg  5886  dmun  5898  dm0rn0  5912  dm0rn0OLD  5913  nfdm  5939  fliftf  7343  opabdm  33274  dmxrn  39510  dmcnvep  39511  rncossdmcoss  39668  dfatco  48529