Theorem fnmap

Description: Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003) (Revised by Mario Carneiro, 8-Sep-2013)

	Ref	Expression
Assertion	fnmap	$$⊢{\uparrow }_{𝑚}Fn\left(\mathrm{V}\times \mathrm{V}\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-map	$$⊢{\uparrow }_{𝑚}=\left(x\in \mathrm{V},y\in \mathrm{V}⟼\left\{f|f:y⟶x\right\}\right)$$
2		mapex	$$⊢\left(y\in \mathrm{V}\wedge x\in \mathrm{V}\right)\to \left\{f|f:y⟶x\right\}\in \mathrm{V}$$
3	2	el2v	$$⊢\left\{f|f:y⟶x\right\}\in \mathrm{V}$$
4	1 3	fnmpoi	$$⊢{\uparrow }_{𝑚}Fn\left(\mathrm{V}\times \mathrm{V}\right)$$