Theorem ulmscl

Description: Closure of the base set in a uniform limit. (Contributed by Mario Carneiro, 26-Feb-2015)

	Ref	Expression
Assertion	ulmscl	$$⊢F\underset{u}{⇝}\left(S\right)G\to S\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		df-br	$$⊢F\underset{u}{⇝}\left(S\right)G\leftrightarrow ⟨F,G⟩\in \underset{u}{⇝}\left(S\right)$$
2		elfvex	$$⊢⟨F,G⟩\in \underset{u}{⇝}\left(S\right)\to S\in \mathrm{V}$$
3	1 2	sylbi	$$⊢F\underset{u}{⇝}\left(S\right)G\to S\in \mathrm{V}$$