Theorem clsneifv4 38409

Description: Value of the the closure (interior) function in terms of the neighborhoods (convergents) function. (Contributed by RP, 27-Jun-2021.)

Hypotheses

Ref	Expression
clsnei.o	|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )
clsnei.p	|- P = ( n e. _V |-> ( p e. ( ~P n ^m ~P n ) |-> ( o e. ~P n |-> ( n \ ( p ` ( n \ o ) ) ) ) ) )
clsnei.d	|- D = ( P ` B )
clsnei.f	|- F = ( ~P B O B )
clsnei.h	|- H = ( F o. D )
clsnei.r	|- ( ph -> K H N )
clsneifv.s	|- ( ph -> S e. ~P B )

Assertion

Ref	Expression
clsneifv4	|- ( ph -> ( K ` S ) = { x e. B | -. ( B \ S ) e. ( N ` x ) } )

Distinct variable groups:   B, i, j, k, l, m, x   B, n, o, p, x   D, i, j, k, l, m   D, n, o, p   i, F, j, k, l   n, F, o, p   i, K, j, k, l, m, x   n, K, o, p   i, N, j, k, l   n, N, o, p   S, m, x   S, o   ph, i, j, k, l, x   ph, n, o, p

Allowed substitution hints:   ph( m)   D( x)   P( x, i, j, k, m, n, o, p, l)   S( i, j, k, n, p, l)   F( x, m)   H( x, i, j, k, m, n, o, p, l)   N( x, m)   O( x, i, j, k, m, n, o, p, l)

Proof of Theorem clsneifv4

Step	Hyp	Ref	Expression
1		dfin5 3582	. 2 |- ( B i^i ( K ` S ) ) = { x e. B | x e. ( K ` S ) }
2		clsnei.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		clsnei.p	. . . . . . 7 |- P = ( n e. _V |-> ( p e. ( ~P n ^m ~P n ) |-> ( o e. ~P n |-> ( n \ ( p ` ( n \ o ) ) ) ) ) )
4		clsnei.d	. . . . . . 7 |- D = ( P ` B )
5		clsnei.f	. . . . . . 7 |- F = ( ~P B O B )
6		clsnei.h	. . . . . . 7 |- H = ( F o. D )
7		clsnei.r	. . . . . . 7 |- ( ph -> K H N )
8	2, 3, 4, 5, 6, 7	clsneikex 38404	. . . . . 6 |- ( ph -> K e. ( ~P B ^m ~P B ) )
9		elmapi 7879	. . . . . 6 |- ( K e. ( ~P B ^m ~P B ) -> K : ~P B --> ~P B )
10	8, 9	syl 17	. . . . 5 |- ( ph -> K : ~P B --> ~P B )
11		clsneifv.s	. . . . 5 |- ( ph -> S e. ~P B )
12	10, 11	ffvelrnd 6360	. . . 4 |- ( ph -> ( K ` S ) e. ~P B )
13	12	elpwid 4170	. . 3 |- ( ph -> ( K ` S ) C_ B )
14		sseqin2 3817	. . 3 |- ( ( K ` S ) C_ B <-> ( B i^i ( K ` S ) ) = ( K ` S ) )
15	13, 14	sylib 208	. 2 |- ( ph -> ( B i^i ( K ` S ) ) = ( K ` S ) )
16	7	adantr 481	. . . 4 |- ( ( ph /\ x e. B ) -> K H N )
17		simpr 477	. . . 4 |- ( ( ph /\ x e. B ) -> x e. B )
18	11	adantr 481	. . . 4 |- ( ( ph /\ x e. B ) -> S e. ~P B )
19	2, 3, 4, 5, 6, 16, 17, 18	clsneiel1 38406	. . 3 |- ( ( ph /\ x e. B ) -> ( x e. ( K ` S ) <-> -. ( B \ S ) e. ( N ` x ) ) )
20	19	rabbidva 3188	. 2 |- ( ph -> { x e. B | x e. ( K ` S ) } = { x e. B | -. ( B \ S ) e. ( N ` x ) } )
21	1, 15, 20	3eqtr3a 2680	1 |- ( ph -> ( K ` S ) = { x e. B | -. ( B \ S ) e. ( N ` x ) } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   o. ccom 5118   -->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: (None)