Theorem ntrneiiex

Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F , then the interior function exists. (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	ntrneiiex	|- ( ph -> I e. ( ~P B ^m ~P 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	ntrneif1o	|- ( ph -> F : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P ~P B ^m B ) )
5		f1orel	|- ( F : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P ~P B ^m B ) -> Rel F )
6	4 5	syl	|- ( ph -> Rel F )
7		releldm	|- ( ( Rel F /\ I F N ) -> I e. dom F )
8	6 3 7	syl2anc	|- ( ph -> I e. dom F )
9		f1odm	|- ( F : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P ~P B ^m B ) -> dom F = ( ~P B ^m ~P B ) )
10	4 9	syl	|- ( ph -> dom F = ( ~P B ^m ~P B ) )
11	8 10	eleqtrd	|- ( ph -> I e. ( ~P B ^m ~P B ) )