Theorem ntrneifv4 38383

Description: The value of the interior (closure) expressed in terms of the neighbors (convergents) function. (Contributed by RP, 26-Jun-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 )
ntrneifv.s	|- ( ph -> S e. ~P B )

Assertion

Ref	Expression
ntrneifv4	|- ( ph -> ( I ` S ) = { x e. B | S e. ( N ` x ) } )

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

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

Proof of Theorem ntrneifv4

Step	Hyp	Ref	Expression
1		dfin5 3582	. 2 |- ( B i^i ( I ` S ) ) = { x e. B | x e. ( I ` S ) }
2		ntrnei.o	. . . . . . 7 |- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )
3		ntrnei.f	. . . . . . 7 |- F = ( ~P B O B )
4		ntrnei.r	. . . . . . 7 |- ( ph -> I F N )
5	2, 3, 4	ntrneiiex 38374	. . . . . 6 |- ( ph -> I e. ( ~P B ^m ~P B ) )
6		elmapi 7879	. . . . . 6 |- ( I e. ( ~P B ^m ~P B ) -> I : ~P B --> ~P B )
7	5, 6	syl 17	. . . . 5 |- ( ph -> I : ~P B --> ~P B )
8		ntrneifv.s	. . . . 5 |- ( ph -> S e. ~P B )
9	7, 8	ffvelrnd 6360	. . . 4 |- ( ph -> ( I ` S ) e. ~P B )
10	9	elpwid 4170	. . 3 |- ( ph -> ( I ` S ) C_ B )
11		sseqin2 3817	. . 3 |- ( ( I ` S ) C_ B <-> ( B i^i ( I ` S ) ) = ( I ` S ) )
12	10, 11	sylib 208	. 2 |- ( ph -> ( B i^i ( I ` S ) ) = ( I ` S ) )
13	4	adantr 481	. . . 4 |- ( ( ph /\ x e. B ) -> I F N )
14		simpr 477	. . . 4 |- ( ( ph /\ x e. B ) -> x e. B )
15	8	adantr 481	. . . 4 |- ( ( ph /\ x e. B ) -> S e. ~P B )
16	2, 3, 13, 14, 15	ntrneiel 38379	. . 3 |- ( ( ph /\ x e. B ) -> ( x e. ( I ` S ) <-> S e. ( N ` x ) ) )
17	16	rabbidva 3188	. 2 |- ( ph -> { x e. B | x e. ( I ` S ) } = { x e. B | S e. ( N ` x ) } )
18	1, 12, 17	3eqtr3a 2680	1 |- ( ph -> ( I ` S ) = { x e. B | S e. ( N ` x ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` 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:  ntrneiel2  38384