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Mirrors > Home > ILE Home > Th. List > df-pm | GIF version |
Description: Define the partial mapping operation. A partial function from 𝐵 to 𝐴 is a function from a subset of 𝐵 to 𝐴. The set of all partial functions from 𝐵 to 𝐴 is written (𝐴 ↑pm 𝐵) (see pmvalg 6659). A notation for this operation apparently does not appear in the literature. We use ↑pm to distinguish it from the less general set exponentiation operation ↑𝑚 (df-map 6650) . See mapsspm 6682 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.) |
Ref | Expression |
---|---|
df-pm | ⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpm 6649 | . 2 class ↑pm | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | cvv 2738 | . . 3 class V | |
5 | vf | . . . . . 6 setvar 𝑓 | |
6 | 5 | cv 1352 | . . . . 5 class 𝑓 |
7 | 6 | wfun 5211 | . . . 4 wff Fun 𝑓 |
8 | 3 | cv 1352 | . . . . . 6 class 𝑦 |
9 | 2 | cv 1352 | . . . . . 6 class 𝑥 |
10 | 8, 9 | cxp 4625 | . . . . 5 class (𝑦 × 𝑥) |
11 | 10 | cpw 3576 | . . . 4 class 𝒫 (𝑦 × 𝑥) |
12 | 7, 5, 11 | crab 2459 | . . 3 class {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} |
13 | 2, 3, 4, 4, 12 | cmpo 5877 | . 2 class (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓}) |
14 | 1, 13 | wceq 1353 | 1 wff ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓}) |
Colors of variables: wff set class |
This definition is referenced by: fnpm 6656 pmvalg 6659 elpmi 6667 pmresg 6676 pmsspw 6683 |
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