Definition df-pm 6003

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 6011). 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 6002) . See mapsspm 6022 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.)

Assertion

Ref	Expression
df-pm	⊢ ↑pm = (x ∈ V, y ∈ V ↦ {f ∈ ℘(y × x) ∣ Fun f})

Distinct variable group:   x,y,f

Detailed syntax breakdown of Definition df-pm

Step	Hyp	Ref	Expression
1		cpm 6001	. 2 class ↑pm
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cvv 2860	. . 3 class V
5		vf	. . . . . 6 setvar f
6	5	cv 1641	. . . . 5 class f
7	6	wfun 4776	. . . 4 wff Fun f
8	3	cv 1641	. . . . . 6 class y
9	2	cv 1641	. . . . . 6 class x
10	8, 9	cxp 4771	. . . . 5 class (y × x)
11	10	cpw 3723	. . . 4 class ℘(y × x)
12	7, 5, 11	crab 2619	. . 3 class {f ∈ ℘(y × x) ∣ Fun f}
13	2, 3, 4, 4, 12	cmpt2 5654	. 2 class (x ∈ V, y ∈ V ↦ {f ∈ ℘(y × x) ∣ Fun f})
14	1, 13	wceq 1642	1 wff ↑pm = (x ∈ V, y ∈ V ↦ {f ∈ ℘(y × x) ∣ Fun f})

This definition is referenced by:  fnpm  6009  pmvalg  6011