Theorem fnpm

Description: Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013)

	Ref	Expression
Assertion	fnpm	⊢ ↑pm Fn ( V × V )

Proof

Step	Hyp	Ref	Expression
1		df-pm	⊢ ↑pm = ( 𝑥 ∈ V , 𝑦 ∈ V ↦ { 𝑓 ∈ 𝒫 ( 𝑦 × 𝑥 ) ∣ Fun 𝑓 } )
2		vex	⊢ 𝑦 ∈ V
3		vex	⊢ 𝑥 ∈ V
4	2 3	xpex	⊢ ( 𝑦 × 𝑥 ) ∈ V
5	4	pwex	⊢ 𝒫 ( 𝑦 × 𝑥 ) ∈ V
6	5	rabex	⊢ { 𝑓 ∈ 𝒫 ( 𝑦 × 𝑥 ) ∣ Fun 𝑓 } ∈ V
7	1 6	fnmpoi	⊢ ↑pm Fn ( V × V )