Metamath Proof Explorer


Theorem uspreg

Description: If a uniform space is Hausdorff, it is regular. Proposition 3 of BourbakiTop1 p. II.5. (Contributed by Thierry Arnoux, 4-Jan-2018)

Ref Expression
Hypothesis uspreg.1
|- J = ( TopOpen ` W )
Assertion uspreg
|- ( ( W e. UnifSp /\ J e. Haus ) -> J e. Reg )

Proof

Step Hyp Ref Expression
1 uspreg.1
 |-  J = ( TopOpen ` W )
2 eqid
 |-  ( Base ` W ) = ( Base ` W )
3 eqid
 |-  ( UnifSt ` W ) = ( UnifSt ` W )
4 2 3 1 isusp
 |-  ( W e. UnifSp <-> ( ( UnifSt ` W ) e. ( UnifOn ` ( Base ` W ) ) /\ J = ( unifTop ` ( UnifSt ` W ) ) ) )
5 4 simprbi
 |-  ( W e. UnifSp -> J = ( unifTop ` ( UnifSt ` W ) ) )
6 5 adantr
 |-  ( ( W e. UnifSp /\ J e. Haus ) -> J = ( unifTop ` ( UnifSt ` W ) ) )
7 4 simplbi
 |-  ( W e. UnifSp -> ( UnifSt ` W ) e. ( UnifOn ` ( Base ` W ) ) )
8 simpr
 |-  ( ( W e. UnifSp /\ J e. Haus ) -> J e. Haus )
9 6 8 eqeltrrd
 |-  ( ( W e. UnifSp /\ J e. Haus ) -> ( unifTop ` ( UnifSt ` W ) ) e. Haus )
10 eqid
 |-  ( unifTop ` ( UnifSt ` W ) ) = ( unifTop ` ( UnifSt ` W ) )
11 10 utopreg
 |-  ( ( ( UnifSt ` W ) e. ( UnifOn ` ( Base ` W ) ) /\ ( unifTop ` ( UnifSt ` W ) ) e. Haus ) -> ( unifTop ` ( UnifSt ` W ) ) e. Reg )
12 7 9 11 syl2an2r
 |-  ( ( W e. UnifSp /\ J e. Haus ) -> ( unifTop ` ( UnifSt ` W ) ) e. Reg )
13 6 12 eqeltrd
 |-  ( ( W e. UnifSp /\ J e. Haus ) -> J e. Reg )