Theorem neipcfilu 20965

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 995	. . . . 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 19604	. . . . 5 |- ( W e. TopSp <-> J e. (TopOn ` X ) )
5	1, 4	sylib 196	. . . 4 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> J e. (TopOn ` X ) )
6		simp3 996	. . . . 5 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> P e. X )
7	6	snssd 4161	. . . 4 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> { P } C_ X )
8		snnzg 4133	. . . . 5 |- ( P e. X -> { P } =/= (/) )
9	6, 8	syl 16	. . . 4 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> { P } =/= (/) )
10		neifil 20547	. . . 4 |- ( ( J e. (TopOn ` X ) /\ { P } C_ X /\ { P } =/= (/) ) -> ( ( nei ` J ) ` { P } ) e. ( Fil ` X ) )
11	5, 7, 9, 10	syl3anc 1226	. . 3 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( nei ` J ) ` { P } ) e. ( Fil ` X ) )
12		filfbas 20515	. . 3 |- ( ( ( nei ` J ) ` { P } ) e. ( Fil ` X ) -> ( ( nei ` J ) ` { P } ) e. ( fBas ` X ) )
13	11, 12	syl 16	. 2 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( nei ` J ) ` { P } ) e. ( fBas ` X ) )
14		eqid 2454	. . . . . . . . . 10 |- ( w " { P } ) = ( w " { P } )
15		imaeq1 5320	. . . . . . . . . . . 12 |- ( v = w -> ( v " { P } ) = ( w " { P } ) )
16	15	eqeq2d 2468	. . . . . . . . . . 11 |- ( v = w -> ( ( w " { P } ) = ( v " { P } ) <-> ( w " { P } ) = ( w " { P } ) ) )
17	16	rspcev 3207	. . . . . . . . . 10 |- ( ( w e. U /\ ( w " { P } ) = ( w " { P } ) ) -> E. v e. U ( w " { P } ) = ( v " { P } ) )
18	14, 17	mpan2 669	. . . . . . . . 9 |- ( w e. U -> E. v e. U ( w " { P } ) = ( v " { P } ) )
19		vex 3109	. . . . . . . . . 10 |- w e. _V
20		imaexg 6710	. . . . . . . . . 10 |- ( w e. _V -> ( w " { P } ) e. _V )
21		eqid 2454	. . . . . . . . . . 11 |- ( v e. U |-> ( v " { P } ) ) = ( v e. U |-> ( v " { P } ) )
22	21	elrnmpt 5238	. . . . . . . . . 10 |- ( ( w " { P } ) e. _V -> ( ( w " { P } ) e. ran ( v e. U |-> ( v " { P } ) ) <-> E. v e. U ( w " { P } ) = ( v " { P } ) ) )
23	19, 20, 22	mp2b 10	. . . . . . . . 9 |- ( ( w " { P } ) e. ran ( v e. U |-> ( v " { P } ) ) <-> E. v e. U ( w " { P } ) = ( v " { P } ) )
24	18, 23	sylibr 212	. . . . . . . 8 |- ( w e. U -> ( w " { P } ) e. ran ( v e. U |-> ( v " { P } ) ) )
25	24	ad2antlr 724	. . . . . . 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 20930	. . . . . . . . . . . 12 |- ( W e. UnifSp <-> ( U e. (UnifOn ` X ) /\ J = (unifTop ` U ) ) )
28	27	simplbi 458	. . . . . . . . . . 11 |- ( W e. UnifSp -> U e. (UnifOn ` X ) )
29	28	3ad2ant1 1015	. . . . . . . . . 10 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> U e. (UnifOn ` X ) )
30		eqid 2454	. . . . . . . . . . 11 |- (unifTop ` U ) = (unifTop ` U )
31	30	utopsnneip 20917	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( ( nei ` (unifTop ` U ) ) ` { P } ) = ran ( v e. U |-> ( v " { P } ) ) )
32	29, 6, 31	syl2anc 659	. . . . . . . . 9 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( nei ` (unifTop ` U ) ) ` { P } ) = ran ( v e. U |-> ( v " { P } ) ) )
33	32	eleq2d 2524	. . . . . . . 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 727	. . . . . . 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 232	. . . . . 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 997	. . . . . . . . . 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 1216	. . . . . . . . 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 462	. . . . . . . . 9 |- ( W e. UnifSp -> J = (unifTop ` U ) )
39	37, 38	syl 16	. . . . . . . 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 5852	. . . . . . 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 5850	. . . . . 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 2545	. . . . 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 459	. . . . 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 5014	. . . . . . 7 |- ( a = ( w " { P } ) -> ( a X. a ) = ( ( w " { P } ) X. ( w " { P } ) ) )
46	45	sseq1d 3516	. . . . . 6 |- ( a = ( w " { P } ) -> ( ( a X. a ) C_ v <-> ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) )
47	46	rspcev 3207	. . . . 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 659	. . . 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 463	. . . . 5 |- ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) -> U e. (UnifOn ` X ) )
50	6	adantr 463	. . . . 5 |- ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) -> P e. X )
51		simpr 459	. . . . 5 |- ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) -> v e. U )
52		simpll1 1033	. . . . . . . 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 753	. . . . . . . 8 |- ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) -> u e. U )
54		ustexsym 20884	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ u e. U ) -> E. w e. U ( `' w = w /\ w C_ u ) )
55	52, 53, 54	syl2anc 659	. . . . . . 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 723	. . . . . . . . . . . 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 753	. . . . . . . . . . . 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 20873	. . . . . . . . . . . 12 |- ( ( U e. (UnifOn ` X ) /\ w e. U ) -> w C_ ( X X. X ) )
59	56, 57, 58	syl2anc 659	. . . . . . . . . . 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 1034	. . . . . . . . . . . 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 1216	. . . . . . . . . . 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 20892	. . . . . . . . . . 11 |- ( ( w C_ ( X X. X ) /\ P e. X ) -> ( ( w " { P } ) X. ( w " { P } ) ) C_ ( w o. `' w ) )
63	59, 61, 62	syl2anc 659	. . . . . . . . . 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 754	. . . . . . . . . . . 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 5154	. . . . . . . . . . 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 5147	. . . . . . . . . . . . . 14 |- ( w C_ u -> ( w o. w ) C_ ( u o. w ) )
67		coss2 5148	. . . . . . . . . . . . . 14 |- ( w C_ u -> ( u o. w ) C_ ( u o. u ) )
68	66, 67	sstrd 3499	. . . . . . . . . . . . 13 |- ( w C_ u -> ( w o. w ) C_ ( u o. u ) )
69	68	ad2antll 726	. . . . . . . . . . . 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 758	. . . . . . . . . . . 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 3499	. . . . . . . . . . 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 3523	. . . . . . . . . 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 3499	. . . . . . . . 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 432	. . . . . . . 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 2929	. . . . . . 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 20879	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ v e. U ) -> E. u e. U ( u o. u ) C_ v )
78	77	3adant2 1013	. . . . . 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 2996	. . . . 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 1226	. . . 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 2996	. . 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 2868	. 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 20957	. . 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 16	. 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 920	1 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( nei ` J ) ` { P } ) e. (CauFilu ` U ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   E.wrex 2805   _Vcvv 3106   C_ wss 3461   (/)c0 3783   {csn 4016   |-> cmpt 4497   X. cxp 4986   `'ccnv 4987   ran crn 4989   "cima 4991   o. ccom 4992   ` cfv 5570   Basecbs 14716   TopOpenctopn 14911   fBascfbas 18601  TopOnctopon 19562   TopSpctps 19564   neicnei 19765   Filcfil 20512  UnifOncust 20868  unifTopcutop 20899  UnifStcuss 20922  UnifSpcusp 20923  CauFil_uccfilu 20955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-fin 7513  df-fi 7863  df-fbas 18611  df-top 19566  df-topon 19569  df-topsp 19570  df-nei 19766  df-fil 20513  df-ust 20869  df-utop 20900  df-usp 20926  df-cfilu 20956

This theorem is referenced by:  ucnextcn  20973