Definition df-map 6002

Description: Define the mapping operation or set exponentiation. The set of all functions that map from B to A is written (A ↑_m B) (see mapval 6012). Many authors write A followed by B as a superscript for this operation and rely on context to avoid confusion other exponentiation operations (e.g. Definition 10.42 of [TakeutiZaring] p. 95). Other authors show B as a prefixed superscript, which is read "A pre B " (e.g. definition of [Enderton] p. 52). Definition 8.21 of [Eisenberg] p. 125 uses the notation Map(B, A) for our (A ↑_m B). The up-arrow is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976). We adopt the first case of his notation (simple exponentiation) and subscript it with m to distinguish it from other kinds of exponentiation. (Contributed by NM, 15-Nov-2007.)

Assertion

Ref	Expression
df-map	⊢ ↑m = (x ∈ V, y ∈ V ↦ {f ∣ f:y–→x})

Distinct variable group:   x,y,f

Detailed syntax breakdown of Definition df-map

Step	Hyp	Ref	Expression
1		cmap 6000	. 2 class ↑m
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cvv 2860	. . 3 class V
5	3	cv 1641	. . . . 5 class y
6	2	cv 1641	. . . . 5 class x
7		vf	. . . . . 6 setvar f
8	7	cv 1641	. . . . 5 class f
9	5, 6, 8	wf 4778	. . . 4 wff f:y–→x
10	9, 7	cab 2339	. . 3 class {f ∣ f:y–→x}
11	2, 3, 4, 4, 10	cmpt2 5654	. 2 class (x ∈ V, y ∈ V ↦ {f ∣ f:y–→x})
12	1, 11	wceq 1642	1 wff ↑m = (x ∈ V, y ∈ V ↦ {f ∣ f:y–→x})

This definition is referenced by:  fnmap  6008  mapvalg  6010