Definition df-dm 4784

Description: Define the domain of a class. Definition 3 of [Suppes] p. 59. For example, F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } → dom F = { 2 , 3 } . Contrast with range (defined in df-rn 4785). For alternate definitions see dfdm2 5322, dfdm3 4967, and dfdm4 4973. 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 4774	. 2 class dom 𝐴
3		vx	. . . . . 6 setvar 𝑥
4	3	cv 1401	. . . . 5 class 𝑥
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1401	. . . . 5 class 𝑦
7	4, 6, 1	wbr 4130	. . . 4 wff 𝑥𝐴𝑦
8	7, 5	wex 1545	. . 3 wff ∃𝑦 𝑥𝐴𝑦
9	8, 3	cab 2224	. 2 class {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
10	2, 9	wceq 1402	1 wff dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}

This definition is used by:  dfdm3  4967  dfrn2  4968  dfdm4  4973  dfdmf  4974  eldmg  4976  dmun  4988  dm0rn0  4998  dmmrnm  5001  nfdm  5026  fliftf  6005