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Metamath Proof Explorer

Theorem utopsnnei

Description: Images of singletons by entourages V are neighborhoods of those singletons. (Contributed by Thierry Arnoux, 13-Jan-2018)

Ref Expression
Hypothesis utoptop.1 
|- J = ( unifTop ` U )
Assertion utopsnnei 
|- ( ( U e. ( UnifOn ` X ) /\ V e. U /\ P e. X ) -> ( V " { P } ) e. ( ( nei ` J ) ` { P } ) )

Proof

Step Hyp Ref Expression
1 utoptop.1 
 |-  J = ( unifTop ` U )
2 eqid 
 |-  ( V " { P } ) = ( V " { P } )
3 imaeq1 
 |-  ( v = V -> ( v " { P } ) = ( V " { P } ) )
4 3 rspceeqv 
 |-  ( ( V e. U /\ ( V " { P } ) = ( V " { P } ) ) -> E. v e. U ( V " { P } ) = ( v " { P } ) )
5 2 4 mpan2 
 |-  ( V e. U -> E. v e. U ( V " { P } ) = ( v " { P } ) )
6 5 3ad2ant2 
 |-  ( ( U e. ( UnifOn ` X ) /\ V e. U /\ P e. X ) -> E. v e. U ( V " { P } ) = ( v " { P } ) )
7 1 utopsnneip 
 |-  ( ( U e. ( UnifOn ` X ) /\ P e. X ) -> ( ( nei ` J ) ` { P } ) = ran ( v e. U |-> ( v " { P } ) ) )
8 7 3adant2 
 |-  ( ( U e. ( UnifOn ` X ) /\ V e. U /\ P e. X ) -> ( ( nei ` J ) ` { P } ) = ran ( v e. U |-> ( v " { P } ) ) )
9 8 eleq2d 
 |-  ( ( U e. ( UnifOn ` X ) /\ V e. U /\ P e. X ) -> ( ( V " { P } ) e. ( ( nei ` J ) ` { P } ) <-> ( V " { P } ) e. ran ( v e. U |-> ( v " { P } ) ) ) )
10 imaexg 
 |-  ( V e. U -> ( V " { P } ) e. _V )
11 eqid 
 |-  ( v e. U |-> ( v " { P } ) ) = ( v e. U |-> ( v " { P } ) )
12 11 elrnmpt 
 |-  ( ( V " { P } ) e. _V -> ( ( V " { P } ) e. ran ( v e. U |-> ( v " { P } ) ) <-> E. v e. U ( V " { P } ) = ( v " { P } ) ) )
13 10 12 syl 
 |-  ( V e. U -> ( ( V " { P } ) e. ran ( v e. U |-> ( v " { P } ) ) <-> E. v e. U ( V " { P } ) = ( v " { P } ) ) )
14 13 3ad2ant2 
 |-  ( ( U e. ( UnifOn ` X ) /\ V e. U /\ P e. X ) -> ( ( V " { P } ) e. ran ( v e. U |-> ( v " { P } ) ) <-> E. v e. U ( V " { P } ) = ( v " { P } ) ) )
15 9 14 bitrd 
 |-  ( ( U e. ( UnifOn ` X ) /\ V e. U /\ P e. X ) -> ( ( V " { P } ) e. ( ( nei ` J ) ` { P } ) <-> E. v e. U ( V " { P } ) = ( v " { P } ) ) )
16 6 15 mpbird 
 |-  ( ( U e. ( UnifOn ` X ) /\ V e. U /\ P e. X ) -> ( V " { P } ) e. ( ( nei ` J ) ` { P } ) )