Theorem tusunif

Description: The uniform structure of a constructed uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017)

	Ref	Expression
Hypothesis	tuslem.k	⊢ 𝐾 = ( toUnifSp ‘ 𝑈 )
Assertion	tusunif	⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑈 = ( UnifSet ‘ 𝐾 ) )

Proof

Step	Hyp	Ref	Expression
1		tuslem.k	⊢ 𝐾 = ( toUnifSp ‘ 𝑈 )
2	1	tuslem	⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( 𝑋 = ( Base ‘ 𝐾 ) ∧ 𝑈 = ( UnifSet ‘ 𝐾 ) ∧ ( unifTop ‘ 𝑈 ) = ( TopOpen ‘ 𝐾 ) ) )
3	2	simp2d	⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑈 = ( UnifSet ‘ 𝐾 ) )