Theorem ntrneineine0

Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F , then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 29-May-2021)

	Ref	Expression
Hypotheses	ntrnei.o	|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )
	ntrnei.f	|- F = ( ~P B O B )
	ntrnei.r	|- ( ph -> I F N )
Assertion	ntrneineine0	|- ( ph -> ( A. x e. B E. s e. ~P B x e. ( I ` s ) <-> A. x e. B ( N ` x ) =/= (/) ) )

Proof

Step	Hyp	Ref	Expression
1		ntrnei.o	|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )
2		ntrnei.f	|- F = ( ~P B O B )
3		ntrnei.r	|- ( ph -> I F N )
4	3	adantr	|- ( ( ph /\ x e. B ) -> I F N )
5		simpr	|- ( ( ph /\ x e. B ) -> x e. B )
6	1 2 4 5	ntrneineine0lem	|- ( ( ph /\ x e. B ) -> ( E. s e. ~P B x e. ( I ` s ) <-> ( N ` x ) =/= (/) ) )
7	6	ralbidva	|- ( ph -> ( A. x e. B E. s e. ~P B x e. ( I ` s ) <-> A. x e. B ( N ` x ) =/= (/) ) )