Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-pm Structured version   Visualization version   GIF version

Definition df-pm 7857
 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 7865). 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 7856) . See mapsspm 7888 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.)
Assertion
Ref Expression
df-pm pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
Distinct variable group:   𝑥,𝑦,𝑓

Detailed syntax breakdown of Definition df-pm
StepHypRef Expression
1 cpm 7855 . 2 class pm
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cvv 3198 . . 3 class V
5 vf . . . . . 6 setvar 𝑓
65cv 1481 . . . . 5 class 𝑓
76wfun 5880 . . . 4 wff Fun 𝑓
83cv 1481 . . . . . 6 class 𝑦
92cv 1481 . . . . . 6 class 𝑥
108, 9cxp 5110 . . . . 5 class (𝑦 × 𝑥)
1110cpw 4156 . . . 4 class 𝒫 (𝑦 × 𝑥)
127, 5, 11crab 2915 . . 3 class {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓}
132, 3, 4, 4, 12cmpt2 6649 . 2 class (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
141, 13wceq 1482 1 wff pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
 Colors of variables: wff setvar class This definition is referenced by:  fnpm  7862  pmvalg  7865
 Copyright terms: Public domain W3C validator