Definition df-dm 5565

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 28218). Another example is the domain of the complex arctangent, (𝐴 ∈ dom arctan ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ -i ∧ 𝐴 ≠ i)) (for proof see atandm 25454). Contrast with range (defined in df-rn 5566). For alternate definitions see dfdm2 6132, dfdm3 5758, and dfdm4 5764. 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 5555	. 2 class dom 𝐴
3		vx	. . . . . 6 setvar 𝑥
4	3	cv 1536	. . . . 5 class 𝑥
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1536	. . . . 5 class 𝑦
7	4, 6, 1	wbr 5066	. . . 4 wff 𝑥𝐴𝑦
8	7, 5	wex 1780	. . 3 wff ∃𝑦 𝑥𝐴𝑦
9	8, 3	cab 2799	. 2 class {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
10	2, 9	wceq 1537	1 wff dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}

This definition is referenced by:  dfdm3  5758  dfrn2  5759  dfdm4  5764  dfdmf  5765  eldmg  5767  dmun  5779  domepOLD  5794  dm0rn0  5795  nfdm  5823  fliftf  7068  opabdm  30362  rncossdmcoss  35710  dfatco  43504