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Metamath Proof Explorer

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 )