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Mirrors > Home > ILE Home > Th. List > df-dm | GIF version |
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 4550). For alternate definitions see dfdm2 5073, dfdm3 4726, and dfdm4 4731. The notation "dom " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
df-dm | ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | 1 | cdm 4539 | . 2 class dom 𝐴 |
3 | vx | . . . . . 6 setvar 𝑥 | |
4 | 3 | cv 1330 | . . . . 5 class 𝑥 |
5 | vy | . . . . . 6 setvar 𝑦 | |
6 | 5 | cv 1330 | . . . . 5 class 𝑦 |
7 | 4, 6, 1 | wbr 3929 | . . . 4 wff 𝑥𝐴𝑦 |
8 | 7, 5 | wex 1468 | . . 3 wff ∃𝑦 𝑥𝐴𝑦 |
9 | 8, 3 | cab 2125 | . 2 class {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
10 | 2, 9 | wceq 1331 | 1 wff dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
Colors of variables: wff set class |
This definition is referenced by: dfdm3 4726 dfrn2 4727 dfdm4 4731 dfdmf 4732 eldmg 4734 dmun 4746 dm0rn0 4756 dmmrnm 4758 nfdm 4783 fliftf 5700 |
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