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	⊢ ↑m Fn ( V × V )

Proof

Step	Hyp	Ref	Expression
1		df-map	⊢ ↑m = ( 𝑥 ∈ V , 𝑦 ∈ V ↦ { 𝑓 ∣ 𝑓 : 𝑦 ⟶ 𝑥 } )
2		mapex	⊢ ( ( 𝑦 ∈ V ∧ 𝑥 ∈ V ) → { 𝑓 ∣ 𝑓 : 𝑦 ⟶ 𝑥 } ∈ V )
3	2	el2v	⊢ { 𝑓 ∣ 𝑓 : 𝑦 ⟶ 𝑥 } ∈ V
4	1 3	fnmpoi	⊢ ↑m Fn ( V × V )