Theorem funmpt

Description: A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013)

	Ref	Expression
Assertion	funmpt	$$⊢\mathrm{Fun}\left(x\in A⟼B\right)$$

Proof

Step	Hyp	Ref	Expression
1		funopab4	$$⊢\mathrm{Fun}\left\{⟨x,y⟩|\left(x\in A\wedge y=B\right)\right\}$$
2		df-mpt	$$⊢\left(x\in A⟼B\right)=\left\{⟨x,y⟩|\left(x\in A\wedge y=B\right)\right\}$$
3	2	funeqi	$$⊢\mathrm{Fun}\left(x\in A⟼B\right)\leftrightarrow \mathrm{Fun}\left\{⟨x,y⟩|\left(x\in A\wedge y=B\right)\right\}$$
4	1 3	mpbir	$$⊢\mathrm{Fun}\left(x\in A⟼B\right)$$