Theorem mapsspw

Description: Set exponentiation is a subset of the power set of the Cartesian product of its arguments. (Contributed by NM, 8-Dec-2006) (Revised by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Assertion	mapsspw	$$⊢A^B\subseteq 𝒫\left(B\times A\right)$$

Proof

Step	Hyp	Ref	Expression
1		mapsspm	$$⊢A^B\subseteq A{\uparrow }_{𝑝𝑚}B$$
2		pmsspw	$$⊢A{\uparrow }_{𝑝𝑚}B\subseteq 𝒫\left(B\times A\right)$$
3	1 2	sstri	$$⊢A^B\subseteq 𝒫\left(B\times A\right)$$