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 e. _V , y e. _V |-> { f e. ~P ( y X. 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. x )
11	10	cpw 3558	. . . 4 class ~P ( y X. x )
12	7, 5, 11	crab 2447	. . 3 class { f e. ~P ( y X. x ) | Fun f }
13	2, 3, 4, 4, 12	cmpo 5843	. 2 class ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )
14	1, 13	wceq 1343	1 wff ^pm = ( x e. _V , y e. _V |-> { f e. ~P ( y X. x ) | Fun f } )

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