Metamath Proof Explorer


Definition df-dm

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 ). Another example is the domain of the complex arctangent, ( A e. dom arctan <-> ( A e. CC /\ A =/= -ui /\ A =/= i ) ) (for proof see atandm ). Contrast with range (defined in df-rn ). For alternate definitions see dfdm2 , dfdm3 , and dfdm4 . The notation " dom " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994)

Ref Expression
Assertion df-dm dom 𝐴 = { 𝑥 ∣ ∃ 𝑦 𝑥 𝐴 𝑦 }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA 𝐴
1 0 cdm dom 𝐴
2 vx 𝑥
3 vy 𝑦
4 2 cv 𝑥
5 3 cv 𝑦
6 4 5 0 wbr 𝑥 𝐴 𝑦
7 6 3 wex 𝑦 𝑥 𝐴 𝑦
8 7 2 cab { 𝑥 ∣ ∃ 𝑦 𝑥 𝐴 𝑦 }
9 1 8 wceq dom 𝐴 = { 𝑥 ∣ ∃ 𝑦 𝑥 𝐴 𝑦 }