Theorem ntrneif1o

Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F , we may characterize the relation as part of a 1-to-1 onto function. (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	ntrneif1o	|- ( ph -> F : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P ~P B ^m B ) )

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	1 2 3	ntrneibex	|- ( ph -> B e. _V )
5	4	pwexd	|- ( ph -> ~P B e. _V )
6	1 5 4 2	fsovf1od	|- ( ph -> F : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P ~P B ^m B ) )