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 e. ( UnifOn ` X ) -> U = ( UnifSet ` K ) )

Proof

Step Hyp Ref Expression
1 tuslem.k
 |-  K = ( toUnifSp ` U )
2 1 tuslem
 |-  ( U e. ( UnifOn ` X ) -> ( X = ( Base ` K ) /\ U = ( UnifSet ` K ) /\ ( unifTop ` U ) = ( TopOpen ` K ) ) )
3 2 simp2d
 |-  ( U e. ( UnifOn ` X ) -> U = ( UnifSet ` K ) )