Theorem tustopn

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

	Ref	Expression
Hypotheses	tuslem.k	⊢ 𝐾 = ( toUnifSp ‘ 𝑈 )
	tustopn.j	⊢ 𝐽 = ( unifTop ‘ 𝑈 )
Assertion	tustopn	⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝐽 = ( TopOpen ‘ 𝐾 ) )

Proof

Step	Hyp	Ref	Expression
1		tuslem.k	⊢ 𝐾 = ( toUnifSp ‘ 𝑈 )
2		tustopn.j	⊢ 𝐽 = ( unifTop ‘ 𝑈 )
3	1	tuslem	⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( 𝑋 = ( Base ‘ 𝐾 ) ∧ 𝑈 = ( UnifSet ‘ 𝐾 ) ∧ ( unifTop ‘ 𝑈 ) = ( TopOpen ‘ 𝐾 ) ) )
4	3	simp3d	⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( unifTop ‘ 𝑈 ) = ( TopOpen ‘ 𝐾 ) )
5	2 4	eqtrid	⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝐽 = ( TopOpen ‘ 𝐾 ) )