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 } ) )