Theorem xlimfun

Description: The convergence relation on the extended reals is a function. (Contributed by Glauco Siliprandi, 23-Apr-2023)

	Ref	Expression
Assertion	xlimfun	⊢ Fun ~~>*

Proof

Step	Hyp	Ref	Expression
1		xrhaus	⊢ ( ordTop ‘ ≤ ) ∈ Haus
2		lmfun	⊢ ( ( ordTop ‘ ≤ ) ∈ Haus → Fun ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) )
3	1 2	ax-mp	⊢ Fun ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) )
4		df-xlim	⊢ ~~>* = ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) )
5	4	funeqi	⊢ ( Fun ~~>* ↔ Fun ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) )
6	3 5	mpbir	⊢ Fun ~~>*