Theorem ntrneiel 38379

Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, F, then there is an equivalence between membership in the interior of a set and non-membership in the closure of the complement of the set. (Contributed by RP, 29-May-2021.)

Hypotheses

Ref	Expression
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 )
ntrnei.x	|- ( ph -> X e. B )
ntrnei.s	|- ( ph -> S e. ~P B )

Assertion

Ref	Expression
ntrneiel	|- ( ph -> ( X e. ( I ` S ) <-> S e. ( N ` X ) ) )

Distinct variable groups:   B, i, j, k, l, m   k, I, l, m   S, m   X, l, m   ph, i, j, k, l

Allowed substitution hints:   ph( m)   S( i, j, k, l)   F( i, j, k, m, l)   I( i, j)   N( i, j, k, m, l)   O( i, j, k, m, l)   X( i, j, k)

Proof of Theorem ntrneiel

Step	Hyp	Ref	Expression
1		ntrnei.s	. . 3 |- ( ph -> S e. ~P B )
2		fveq2 6191	. . . . 5 |- ( m = S -> ( I ` m ) = ( I ` S ) )
3	2	eleq2d 2687	. . . 4 |- ( m = S -> ( X e. ( I ` m ) <-> X e. ( I ` S ) ) )
4	3	elrab3 3364	. . 3 |- ( S e. ~P B -> ( S e. { m e. ~P B | X e. ( I ` m ) } <-> X e. ( I ` S ) ) )
5	1, 4	syl 17	. 2 |- ( ph -> ( S e. { m e. ~P B | X e. ( I ` m ) } <-> X e. ( I ` S ) ) )
6		ntrnei.o	. . . . 5 |- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )
7		ntrnei.f	. . . . . . 7 |- F = ( ~P B O B )
8		ntrnei.r	. . . . . . 7 |- ( ph -> I F N )
9	6, 7, 8	ntrneibex 38371	. . . . . 6 |- ( ph -> B e. _V )
10		pwexg 4850	. . . . . 6 |- ( B e. _V -> ~P B e. _V )
11	9, 10	syl 17	. . . . 5 |- ( ph -> ~P B e. _V )
12	6, 7, 8	ntrneiiex 38374	. . . . 5 |- ( ph -> I e. ( ~P B ^m ~P B ) )
13		eqid 2622	. . . . 5 |- ( F ` I ) = ( F ` I )
14		ntrnei.x	. . . . 5 |- ( ph -> X e. B )
15	6, 11, 9, 7, 12, 13, 14	fsovfvfvd 38305	. . . 4 |- ( ph -> ( ( F ` I ) ` X ) = { m e. ~P B | X e. ( I ` m ) } )
16	6, 7, 8	ntrneifv1 38377	. . . . 5 |- ( ph -> ( F ` I ) = N )
17	16	fveq1d 6193	. . . 4 |- ( ph -> ( ( F ` I ) ` X ) = ( N ` X ) )
18	15, 17	eqtr3d 2658	. . 3 |- ( ph -> { m e. ~P B | X e. ( I ` m ) } = ( N ` X ) )
19	18	eleq2d 2687	. 2 |- ( ph -> ( S e. { m e. ~P B | X e. ( I ` m ) } <-> S e. ( N ` X ) ) )
20	5, 19	bitr3d 270	1 |- ( ph -> ( X e. ( I ` S ) <-> S e. ( N ` X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by:  ntrneifv3  38380  ntrneineine0lem  38381  ntrneineine1lem  38382  ntrneifv4  38383  ntrneiel2  38384  ntrneicls00  38387  ntrneicls11  38388  ntrneiiso  38389  ntrneik2  38390  ntrneix2  38391  ntrneikb  38392  ntrneixb  38393  ntrneik3  38394  ntrneix3  38395  ntrneik13  38396  ntrneix13  38397  ntrneik4w  38398  ntrneik4  38399  clsneiel1  38406  neicvgel1  38417