Definition df-map 6612

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 6622). 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, 8-Dec-2003.)

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 6610	. 2 class ↑𝑚
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cvv 2725	. . 3 class V
5	3	cv 1342	. . . . 5 class 𝑦
6	2	cv 1342	. . . . 5 class 𝑥
7		vf	. . . . . 6 setvar 𝑓
8	7	cv 1342	. . . . 5 class 𝑓
9	5, 6, 8	wf 5183	. . . 4 wff 𝑓:𝑦⟶𝑥
10	9, 7	cab 2151	. . 3 class {𝑓 ∣ 𝑓:𝑦⟶𝑥}
11	2, 3, 4, 4, 10	cmpo 5843	. 2 class (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥})
12	1, 11	wceq 1343	1 wff ↑𝑚 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥})

This definition is referenced by:  fnmap  6617  reldmmap  6619  mapvalg  6620  elmapex  6631