Theorem ulmscl

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

	Ref	Expression
Assertion	ulmscl	⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝑆 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		df-br	⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ↔ ⟨ 𝐹 , 𝐺 ⟩ ∈ ( ⇝𝑢 ‘ 𝑆 ) )
2		elfvex	⊢ ( ⟨ 𝐹 , 𝐺 ⟩ ∈ ( ⇝𝑢 ‘ 𝑆 ) → 𝑆 ∈ V )
3	1 2	sylbi	⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝑆 ∈ V )