Definition df-pm 6613

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 6621). 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 6612) . See mapsspm 6644 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

Step	Hyp	Ref	Expression
1		cpm 6611	. 2 class ↑pm
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cvv 2725	. . 3 class V
5		vf	. . . . . 6 setvar 𝑓
6	5	cv 1342	. . . . 5 class 𝑓
7	6	wfun 5181	. . . 4 wff Fun 𝑓
8	3	cv 1342	. . . . . 6 class 𝑦
9	2	cv 1342	. . . . . 6 class 𝑥
10	8, 9	cxp 4601	. . . . 5 class (𝑦 × 𝑥)
11	10	cpw 3558	. . . 4 class 𝒫 (𝑦 × 𝑥)
12	7, 5, 11	crab 2447	. . 3 class {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓}
13	2, 3, 4, 4, 12	cmpo 5843	. 2 class (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
14	1, 13	wceq 1343	1 wff ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})

This definition is referenced by:  fnpm  6618  pmvalg  6621  elpmi  6629  pmresg  6638  pmsspw  6645