Theorem clsneiel1 38406

Description: If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the H operator, then membership in the closure of a subset is equivalent to the complement of the subset not being a neighborhood of the point. (Contributed by RP, 7-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 )
clsneiel.x	|- ( ph -> X e. B )
clsneiel.s	|- ( ph -> S e. ~P B )

Assertion

Ref	Expression
clsneiel1	|- ( ph -> ( X e. ( K ` S ) <-> -. ( B \ S ) e. ( N ` X ) ) )

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

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

Proof of Theorem clsneiel1

Step	Hyp	Ref	Expression
1		clsnei.d	. . . 4 |- D = ( P ` B )
2		clsnei.h	. . . 4 |- H = ( F o. D )
3		clsnei.r	. . . 4 |- ( ph -> K H N )
4	1, 2, 3	clsneibex 38400	. . 3 |- ( ph -> B e. _V )
5	4	ancli 574	. 2 |- ( ph -> ( ph /\ B e. _V ) )
6		clsnei.o	. . . . . 6 |- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )
7		simpr 477	. . . . . . 7 |- ( ( ph /\ B e. _V ) -> B e. _V )
8		pwexg 4850	. . . . . . 7 |- ( B e. _V -> ~P B e. _V )
9	7, 8	syl 17	. . . . . 6 |- ( ( ph /\ B e. _V ) -> ~P B e. _V )
10		clsnei.f	. . . . . 6 |- F = ( ~P B O B )
11	6, 9, 7, 10	fsovfd 38306	. . . . 5 |- ( ( ph /\ B e. _V ) -> F : ( ~P B ^m ~P B ) --> ( ~P ~P B ^m B ) )
12	11	ffnd 6046	. . . 4 |- ( ( ph /\ B e. _V ) -> F Fn ( ~P B ^m ~P B ) )
13		clsnei.p	. . . . . 6 |- P = ( n e. _V |-> ( p e. ( ~P n ^m ~P n ) |-> ( o e. ~P n |-> ( n \ ( p ` ( n \ o ) ) ) ) ) )
14	13, 1, 7	dssmapf1od 38315	. . . . 5 |- ( ( ph /\ B e. _V ) -> D : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P B ^m ~P B ) )
15		f1of 6137	. . . . 5 |- ( D : ( ~P B ^m ~P B ) -1-1-onto-> ( ~P B ^m ~P B ) -> D : ( ~P B ^m ~P B ) --> ( ~P B ^m ~P B ) )
16	14, 15	syl 17	. . . 4 |- ( ( ph /\ B e. _V ) -> D : ( ~P B ^m ~P B ) --> ( ~P B ^m ~P B ) )
17	2	breqi 4659	. . . . . 6 |- ( K H N <-> K ( F o. D ) N )
18	3, 17	sylib 208	. . . . 5 |- ( ph -> K ( F o. D ) N )
19	18	adantr 481	. . . 4 |- ( ( ph /\ B e. _V ) -> K ( F o. D ) N )
20	12, 16, 19	brcoffn 38328	. . 3 |- ( ( ph /\ B e. _V ) -> ( K D ( D ` K ) /\ ( D ` K ) F N ) )
21	20	ancli 574	. 2 |- ( ( ph /\ B e. _V ) -> ( ( ph /\ B e. _V ) /\ ( K D ( D ` K ) /\ ( D ` K ) F N ) ) )
22		simprl 794	. . . 4 |- ( ( ( ph /\ B e. _V ) /\ ( K D ( D ` K ) /\ ( D ` K ) F N ) ) -> K D ( D ` K ) )
23		clsneiel.x	. . . . 5 |- ( ph -> X e. B )
24	23	ad2antrr 762	. . . 4 |- ( ( ( ph /\ B e. _V ) /\ ( K D ( D ` K ) /\ ( D ` K ) F N ) ) -> X e. B )
25		clsneiel.s	. . . . 5 |- ( ph -> S e. ~P B )
26	25	ad2antrr 762	. . . 4 |- ( ( ( ph /\ B e. _V ) /\ ( K D ( D ` K ) /\ ( D ` K ) F N ) ) -> S e. ~P B )
27	13, 1, 22, 24, 26	ntrclselnel1 38355	. . 3 |- ( ( ( ph /\ B e. _V ) /\ ( K D ( D ` K ) /\ ( D ` K ) F N ) ) -> ( X e. ( K ` S ) <-> -. X e. ( ( D ` K ) ` ( B \ S ) ) ) )
28		simprr 796	. . . . 5 |- ( ( ( ph /\ B e. _V ) /\ ( K D ( D ` K ) /\ ( D ` K ) F N ) ) -> ( D ` K ) F N )
29		simplr 792	. . . . . 6 |- ( ( ( ph /\ B e. _V ) /\ ( K D ( D ` K ) /\ ( D ` K ) F N ) ) -> B e. _V )
30		difssd 3738	. . . . . 6 |- ( ( ( ph /\ B e. _V ) /\ ( K D ( D ` K ) /\ ( D ` K ) F N ) ) -> ( B \ S ) C_ B )
31	29, 30	sselpwd 4807	. . . . 5 |- ( ( ( ph /\ B e. _V ) /\ ( K D ( D ` K ) /\ ( D ` K ) F N ) ) -> ( B \ S ) e. ~P B )
32	6, 10, 28, 24, 31	ntrneiel 38379	. . . 4 |- ( ( ( ph /\ B e. _V ) /\ ( K D ( D ` K ) /\ ( D ` K ) F N ) ) -> ( X e. ( ( D ` K ) ` ( B \ S ) ) <-> ( B \ S ) e. ( N ` X ) ) )
33	32	notbid 308	. . 3 |- ( ( ( ph /\ B e. _V ) /\ ( K D ( D ` K ) /\ ( D ` K ) F N ) ) -> ( -. X e. ( ( D ` K ) ` ( B \ S ) ) <-> -. ( B \ S ) e. ( N ` X ) ) )
34	27, 33	bitrd 268	. 2 |- ( ( ( ph /\ B e. _V ) /\ ( K D ( D ` K ) /\ ( D ` K ) F N ) ) -> ( X e. ( K ` S ) <-> -. ( B \ S ) e. ( N ` X ) ) )
35	5, 21, 34	3syl 18	1 |- ( ph -> ( X e. ( K ` S ) <-> -. ( B \ S ) e. ( N ` X ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   \ cdif 3571   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` 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:  clsneiel2  38407  clsneifv4  38409