Metamath Proof Explorer

Theorem ressust

Description: The uniform structure of a restricted space. (Contributed by Thierry Arnoux, 22-Jan-2018)

Ref Expression
Hypotheses ressust.x 
|- X = ( Base ` W )
ressust.t 
|- T = ( UnifSt ` ( W |`s A ) )
Assertion ressust 
|- ( ( W e. UnifSp /\ A C_ X ) -> T e. ( UnifOn ` A ) )

Proof

Step Hyp Ref Expression
1 ressust.x 
 |-  X = ( Base ` W )
2 ressust.t 
 |-  T = ( UnifSt ` ( W |`s A ) )
3 1 fvexi 
 |-  X e. _V
4 3 ssex 
 |-  ( A C_ X -> A e. _V )
5 4 adantl 
 |-  ( ( W e. UnifSp /\ A C_ X ) -> A e. _V )
6 ressuss 
 |-  ( A e. _V -> ( UnifSt ` ( W |`s A ) ) = ( ( UnifSt ` W ) |`t ( A X. A ) ) )
7 5 6 syl 
 |-  ( ( W e. UnifSp /\ A C_ X ) -> ( UnifSt ` ( W |`s A ) ) = ( ( UnifSt ` W ) |`t ( A X. A ) ) )
8 2 7 syl5eq 
 |-  ( ( W e. UnifSp /\ A C_ X ) -> T = ( ( UnifSt ` W ) |`t ( A X. A ) ) )
9 eqid 
 |-  ( UnifSt ` W ) = ( UnifSt ` W )
10 eqid 
 |-  ( TopOpen ` W ) = ( TopOpen ` W )
11 1 9 10 isusp 
 |-  ( W e. UnifSp <-> ( ( UnifSt ` W ) e. ( UnifOn ` X ) /\ ( TopOpen ` W ) = ( unifTop ` ( UnifSt ` W ) ) ) )
12 11 simplbi 
 |-  ( W e. UnifSp -> ( UnifSt ` W ) e. ( UnifOn ` X ) )
13 trust 
 |-  ( ( ( UnifSt ` W ) e. ( UnifOn ` X ) /\ A C_ X ) -> ( ( UnifSt ` W ) |`t ( A X. A ) ) e. ( UnifOn ` A ) )
14 12 13 sylan 
 |-  ( ( W e. UnifSp /\ A C_ X ) -> ( ( UnifSt ` W ) |`t ( A X. A ) ) e. ( UnifOn ` A ) )
15 8 14 eqeltrd 
 |-  ( ( W e. UnifSp /\ A C_ X ) -> T e. ( UnifOn ` A ) )