Metamath Proof Explorer


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 ) )