Metamath Proof Explorer

Theorem ustssxp

Description: Entourages are subsets of the Cartesian product of the base set. (Contributed by Thierry Arnoux, 19-Nov-2017)

Ref Expression
Assertion ustssxp 
|- ( ( U e. ( UnifOn ` X ) /\ V e. U ) -> V C_ ( X X. X ) )

Proof

Step Hyp Ref Expression
1 elfvex 
 |-  ( U e. ( UnifOn ` X ) -> X e. _V )
2 isust 
 |-  ( X e. _V -> ( U e. ( UnifOn ` X ) <-> ( U C_ ~P ( X X. X ) /\ ( X X. X ) e. U /\ A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) ) ) )
3 1 2 syl 
 |-  ( U e. ( UnifOn ` X ) -> ( U e. ( UnifOn ` X ) <-> ( U C_ ~P ( X X. X ) /\ ( X X. X ) e. U /\ A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) ) ) )
4 3 ibi 
 |-  ( U e. ( UnifOn ` X ) -> ( U C_ ~P ( X X. X ) /\ ( X X. X ) e. U /\ A. v e. U ( A. w e. ~P ( X X. X ) ( v C_ w -> w e. U ) /\ A. w e. U ( v i^i w ) e. U /\ ( ( _I |` X ) C_ v /\ `' v e. U /\ E. w e. U ( w o. w ) C_ v ) ) ) )
5 4 simp1d 
 |-  ( U e. ( UnifOn ` X ) -> U C_ ~P ( X X. X ) )
6 5 sselda 
 |-  ( ( U e. ( UnifOn ` X ) /\ V e. U ) -> V e. ~P ( X X. X ) )
7 6 elpwid 
 |-  ( ( U e. ( UnifOn ` X ) /\ V e. U ) -> V C_ ( X X. X ) )