Theorem neipcfilu 22100

Description: In an uniform space, a neighboring filter is a Cauchy filter base. (Contributed by Thierry Arnoux, 24-Jan-2018.)

Hypotheses

Ref	Expression
neipcfilu.x	|- X = ( Base ` W )
neipcfilu.j	|- J = ( TopOpen ` W )
neipcfilu.u	|- U = (UnifSt ` W )

Assertion

Ref	Expression
neipcfilu	|- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( nei ` J ) ` { P } ) e. (CauFilu ` U ) )

Proof of Theorem neipcfilu

Dummy variables v a w u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1062	. . . . 5 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> W e. TopSp )
2		neipcfilu.x	. . . . . 6 |- X = ( Base ` W )
3		neipcfilu.j	. . . . . 6 |- J = ( TopOpen ` W )
4	2, 3	istps 20738	. . . . 5 |- ( W e. TopSp <-> J e. (TopOn ` X ) )
5	1, 4	sylib 208	. . . 4 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> J e. (TopOn ` X ) )
6		simp3 1063	. . . . 5 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> P e. X )
7	6	snssd 4340	. . . 4 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> { P } C_ X )
8		snnzg 4308	. . . . 5 |- ( P e. X -> { P } =/= (/) )
9	6, 8	syl 17	. . . 4 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> { P } =/= (/) )
10		neifil 21684	. . . 4 |- ( ( J e. (TopOn ` X ) /\ { P } C_ X /\ { P } =/= (/) ) -> ( ( nei ` J ) ` { P } ) e. ( Fil ` X ) )
11	5, 7, 9, 10	syl3anc 1326	. . 3 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( nei ` J ) ` { P } ) e. ( Fil ` X ) )
12		filfbas 21652	. . 3 |- ( ( ( nei ` J ) ` { P } ) e. ( Fil ` X ) -> ( ( nei ` J ) ` { P } ) e. ( fBas ` X ) )
13	11, 12	syl 17	. 2 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( nei ` J ) ` { P } ) e. ( fBas ` X ) )
14		eqid 2622	. . . . . . . . . 10 |- ( w " { P } ) = ( w " { P } )
15		imaeq1 5461	. . . . . . . . . . . 12 |- ( v = w -> ( v " { P } ) = ( w " { P } ) )
16	15	eqeq2d 2632	. . . . . . . . . . 11 |- ( v = w -> ( ( w " { P } ) = ( v " { P } ) <-> ( w " { P } ) = ( w " { P } ) ) )
17	16	rspcev 3309	. . . . . . . . . 10 |- ( ( w e. U /\ ( w " { P } ) = ( w " { P } ) ) -> E. v e. U ( w " { P } ) = ( v " { P } ) )
18	14, 17	mpan2 707	. . . . . . . . 9 |- ( w e. U -> E. v e. U ( w " { P } ) = ( v " { P } ) )
19		vex 3203	. . . . . . . . . . 11 |- w e. _V
20	19	imaex 7104	. . . . . . . . . 10 |- ( w " { P } ) e. _V
21		eqid 2622	. . . . . . . . . . 11 |- ( v e. U |-> ( v " { P } ) ) = ( v e. U |-> ( v " { P } ) )
22	21	elrnmpt 5372	. . . . . . . . . 10 |- ( ( w " { P } ) e. _V -> ( ( w " { P } ) e. ran ( v e. U |-> ( v " { P } ) ) <-> E. v e. U ( w " { P } ) = ( v " { P } ) ) )
23	20, 22	ax-mp 5	. . . . . . . . 9 |- ( ( w " { P } ) e. ran ( v e. U |-> ( v " { P } ) ) <-> E. v e. U ( w " { P } ) = ( v " { P } ) )
24	18, 23	sylibr 224	. . . . . . . 8 |- ( w e. U -> ( w " { P } ) e. ran ( v e. U |-> ( v " { P } ) ) )
25	24	ad2antlr 763	. . . . . . 7 |- ( ( ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) /\ w e. U ) /\ ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) -> ( w " { P } ) e. ran ( v e. U |-> ( v " { P } ) ) )
26		neipcfilu.u	. . . . . . . . . . . . 13 |- U = (UnifSt ` W )
27	2, 26, 3	isusp 22065	. . . . . . . . . . . 12 |- ( W e. UnifSp <-> ( U e. (UnifOn ` X ) /\ J = (unifTop ` U ) ) )
28	27	simplbi 476	. . . . . . . . . . 11 |- ( W e. UnifSp -> U e. (UnifOn ` X ) )
29	28	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> U e. (UnifOn ` X ) )
30		eqid 2622	. . . . . . . . . . 11 |- (unifTop ` U ) = (unifTop ` U )
31	30	utopsnneip 22052	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( ( nei ` (unifTop ` U ) ) ` { P } ) = ran ( v e. U |-> ( v " { P } ) ) )
32	29, 6, 31	syl2anc 693	. . . . . . . . 9 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( nei ` (unifTop ` U ) ) ` { P } ) = ran ( v e. U |-> ( v " { P } ) ) )
33	32	eleq2d 2687	. . . . . . . 8 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( w " { P } ) e. ( ( nei ` (unifTop ` U ) ) ` { P } ) <-> ( w " { P } ) e. ran ( v e. U |-> ( v " { P } ) ) ) )
34	33	ad3antrrr 766	. . . . . . 7 |- ( ( ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) /\ w e. U ) /\ ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) -> ( ( w " { P } ) e. ( ( nei ` (unifTop ` U ) ) ` { P } ) <-> ( w " { P } ) e. ran ( v e. U |-> ( v " { P } ) ) ) )
35	25, 34	mpbird 247	. . . . . 6 |- ( ( ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) /\ w e. U ) /\ ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) -> ( w " { P } ) e. ( ( nei ` (unifTop ` U ) ) ` { P } ) )
36		simpl1 1064	. . . . . . . . . 10 |- ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ ( v e. U /\ w e. U /\ ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) ) -> W e. UnifSp )
37	36	3anassrs 1290	. . . . . . . . 9 |- ( ( ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) /\ w e. U ) /\ ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) -> W e. UnifSp )
38	27	simprbi 480	. . . . . . . . 9 |- ( W e. UnifSp -> J = (unifTop ` U ) )
39	37, 38	syl 17	. . . . . . . 8 |- ( ( ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) /\ w e. U ) /\ ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) -> J = (unifTop ` U ) )
40	39	fveq2d 6195	. . . . . . 7 |- ( ( ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) /\ w e. U ) /\ ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) -> ( nei ` J ) = ( nei ` (unifTop ` U ) ) )
41	40	fveq1d 6193	. . . . . 6 |- ( ( ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) /\ w e. U ) /\ ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) -> ( ( nei ` J ) ` { P } ) = ( ( nei ` (unifTop ` U ) ) ` { P } ) )
42	35, 41	eleqtrrd 2704	. . . . 5 |- ( ( ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) /\ w e. U ) /\ ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) -> ( w " { P } ) e. ( ( nei ` J ) ` { P } ) )
43		simpr 477	. . . . 5 |- ( ( ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) /\ w e. U ) /\ ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) -> ( ( w " { P } ) X. ( w " { P } ) ) C_ v )
44		id 22	. . . . . . . 8 |- ( a = ( w " { P } ) -> a = ( w " { P } ) )
45	44	sqxpeqd 5141	. . . . . . 7 |- ( a = ( w " { P } ) -> ( a X. a ) = ( ( w " { P } ) X. ( w " { P } ) ) )
46	45	sseq1d 3632	. . . . . 6 |- ( a = ( w " { P } ) -> ( ( a X. a ) C_ v <-> ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) )
47	46	rspcev 3309	. . . . 5 |- ( ( ( w " { P } ) e. ( ( nei ` J ) ` { P } ) /\ ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) -> E. a e. ( ( nei ` J ) ` { P } ) ( a X. a ) C_ v )
48	42, 43, 47	syl2anc 693	. . . 4 |- ( ( ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) /\ w e. U ) /\ ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) -> E. a e. ( ( nei ` J ) ` { P } ) ( a X. a ) C_ v )
49	29	adantr 481	. . . . 5 |- ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) -> U e. (UnifOn ` X ) )
50	6	adantr 481	. . . . 5 |- ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) -> P e. X )
51		simpr 477	. . . . 5 |- ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) -> v e. U )
52		simpll1 1100	. . . . . . . 8 |- ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) -> U e. (UnifOn ` X ) )
53		simplr 792	. . . . . . . 8 |- ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) -> u e. U )
54		ustexsym 22019	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ u e. U ) -> E. w e. U ( `' w = w /\ w C_ u ) )
55	52, 53, 54	syl2anc 693	. . . . . . 7 |- ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) -> E. w e. U ( `' w = w /\ w C_ u ) )
56	52	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) /\ w e. U ) /\ ( `' w = w /\ w C_ u ) ) -> U e. (UnifOn ` X ) )
57		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) /\ w e. U ) /\ ( `' w = w /\ w C_ u ) ) -> w e. U )
58		ustssxp 22008	. . . . . . . . . . . 12 |- ( ( U e. (UnifOn ` X ) /\ w e. U ) -> w C_ ( X X. X ) )
59	56, 57, 58	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) /\ w e. U ) /\ ( `' w = w /\ w C_ u ) ) -> w C_ ( X X. X ) )
60		simpll2 1101	. . . . . . . . . . . 12 |- ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( ( u o. u ) C_ v /\ w e. U /\ ( `' w = w /\ w C_ u ) ) ) -> P e. X )
61	60	3anassrs 1290	. . . . . . . . . . 11 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) /\ w e. U ) /\ ( `' w = w /\ w C_ u ) ) -> P e. X )
62		ustneism 22027	. . . . . . . . . . 11 |- ( ( w C_ ( X X. X ) /\ P e. X ) -> ( ( w " { P } ) X. ( w " { P } ) ) C_ ( w o. `' w ) )
63	59, 61, 62	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) /\ w e. U ) /\ ( `' w = w /\ w C_ u ) ) -> ( ( w " { P } ) X. ( w " { P } ) ) C_ ( w o. `' w ) )
64		simprl 794	. . . . . . . . . . . 12 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) /\ w e. U ) /\ ( `' w = w /\ w C_ u ) ) -> `' w = w )
65	64	coeq2d 5284	. . . . . . . . . . 11 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) /\ w e. U ) /\ ( `' w = w /\ w C_ u ) ) -> ( w o. `' w ) = ( w o. w ) )
66		coss1 5277	. . . . . . . . . . . . . 14 |- ( w C_ u -> ( w o. w ) C_ ( u o. w ) )
67		coss2 5278	. . . . . . . . . . . . . 14 |- ( w C_ u -> ( u o. w ) C_ ( u o. u ) )
68	66, 67	sstrd 3613	. . . . . . . . . . . . 13 |- ( w C_ u -> ( w o. w ) C_ ( u o. u ) )
69	68	ad2antll 765	. . . . . . . . . . . 12 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) /\ w e. U ) /\ ( `' w = w /\ w C_ u ) ) -> ( w o. w ) C_ ( u o. u ) )
70		simpllr 799	. . . . . . . . . . . 12 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) /\ w e. U ) /\ ( `' w = w /\ w C_ u ) ) -> ( u o. u ) C_ v )
71	69, 70	sstrd 3613	. . . . . . . . . . 11 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) /\ w e. U ) /\ ( `' w = w /\ w C_ u ) ) -> ( w o. w ) C_ v )
72	65, 71	eqsstrd 3639	. . . . . . . . . 10 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) /\ w e. U ) /\ ( `' w = w /\ w C_ u ) ) -> ( w o. `' w ) C_ v )
73	63, 72	sstrd 3613	. . . . . . . . 9 |- ( ( ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) /\ w e. U ) /\ ( `' w = w /\ w C_ u ) ) -> ( ( w " { P } ) X. ( w " { P } ) ) C_ v )
74	73	ex 450	. . . . . . . 8 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) /\ w e. U ) -> ( ( `' w = w /\ w C_ u ) -> ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) )
75	74	reximdva 3017	. . . . . . 7 |- ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) -> ( E. w e. U ( `' w = w /\ w C_ u ) -> E. w e. U ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) )
76	55, 75	mpd 15	. . . . . 6 |- ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) -> E. w e. U ( ( w " { P } ) X. ( w " { P } ) ) C_ v )
77		ustexhalf 22014	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ v e. U ) -> E. u e. U ( u o. u ) C_ v )
78	77	3adant2 1080	. . . . . 6 |- ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) -> E. u e. U ( u o. u ) C_ v )
79	76, 78	r19.29a 3078	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) -> E. w e. U ( ( w " { P } ) X. ( w " { P } ) ) C_ v )
80	49, 50, 51, 79	syl3anc 1326	. . . 4 |- ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) -> E. w e. U ( ( w " { P } ) X. ( w " { P } ) ) C_ v )
81	48, 80	r19.29a 3078	. . 3 |- ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) -> E. a e. ( ( nei ` J ) ` { P } ) ( a X. a ) C_ v )
82	81	ralrimiva 2966	. 2 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> A. v e. U E. a e. ( ( nei ` J ) ` { P } ) ( a X. a ) C_ v )
83		iscfilu 22092	. . 3 |- ( U e. (UnifOn ` X ) -> ( ( ( nei ` J ) ` { P } ) e. (CauFilu ` U ) <-> ( ( ( nei ` J ) ` { P } ) e. ( fBas ` X ) /\ A. v e. U E. a e. ( ( nei ` J ) ` { P } ) ( a X. a ) C_ v ) ) )
84	29, 83	syl 17	. 2 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( ( nei ` J ) ` { P } ) e. (CauFilu ` U ) <-> ( ( ( nei ` J ) ` { P } ) e. ( fBas ` X ) /\ A. v e. U E. a e. ( ( nei ` J ) ` { P } ) ( a X. a ) C_ v ) ) )
85	13, 82, 84	mpbir2and 957	1 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( nei ` J ) ` { P } ) e. (CauFilu ` U ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {csn 4177   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ran crn 5115   "cima 5117   o. ccom 5118   ` cfv 5888   Basecbs 15857   TopOpenctopn 16082   fBascfbas 19734  TopOnctopon 20715   TopSpctps 20736   neicnei 20901   Filcfil 21649  UnifOncust 22003  unifTopcutop 22034  UnifStcuss 22057  UnifSpcusp 22058  CauFil_uccfilu 22090

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-3or 1038  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-fbas 19743  df-top 20699  df-topon 20716  df-topsp 20737  df-nei 20902  df-fil 21650  df-ust 22004  df-utop 22035  df-usp 22061  df-cfilu 22091

This theorem is referenced by:  ucnextcn  22108