Definition df-dm 4786

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 4787). For alternate definitions see dfdm2 5325, dfdm3 4969, and dfdm4 4975. 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 4776	. 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 4131	. . . 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  4969  dfrn2  4970  dfdm4  4975  dfdmf  4976  eldmg  4978  dmun  4990  dm0rn0  5000  dmmrnm  5003  nfdm  5028  fliftf  6008