Theorem mapval

Description: The value of set exponentiation (inference version). ( A ^m B ) is the set of all functions that map from B to A . Definition 10.24 of Kunen p. 24. (Contributed by NM, 8-Dec-2003)

	Ref	Expression
Hypotheses	mapval.1	$$⊢A\in \mathrm{V}$$
	mapval.2	$$⊢B\in \mathrm{V}$$
Assertion	mapval	$$⊢A^B=\left\{f|f:B⟶A\right\}$$

Proof

Step	Hyp	Ref	Expression
1		mapval.1	$$⊢A\in \mathrm{V}$$
2		mapval.2	$$⊢B\in \mathrm{V}$$
3		mapvalg	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to A^B=\left\{f|f:B⟶A\right\}$$
4	1 2 3	mp2an	$$⊢A^B=\left\{f|f:B⟶A\right\}$$