Theorem neipcfilu 21322

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 1010	. . . . 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 19962	. . . . 5 |- ( W e. TopSp <-> J e. (TopOn ` X ) )
5	1, 4	sylib 201	. . . 4 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> J e. (TopOn ` X ) )
6		simp3 1011	. . . . 5 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> P e. X )
7	6	snssd 4086	. . . 4 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> { P } C_ X )
8		snnzg 4058	. . . . 5 |- ( P e. X -> { P } =/= (/) )
9	6, 8	syl 17	. . . 4 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> { P } =/= (/) )
10		neifil 20906	. . . 4 |- ( ( J e. (TopOn ` X ) /\ { P } C_ X /\ { P } =/= (/) ) -> ( ( nei ` J ) ` { P } ) e. ( Fil ` X ) )
11	5, 7, 9, 10	syl3anc 1271	. . 3 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( nei ` J ) ` { P } ) e. ( Fil ` X ) )
12		filfbas 20874	. . 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 2452	. . . . . . . . . 10 |- ( w " { P } ) = ( w " { P } )
15		imaeq1 5141	. . . . . . . . . . . 12 |- ( v = w -> ( v " { P } ) = ( w " { P } ) )
16	15	eqeq2d 2462	. . . . . . . . . . 11 |- ( v = w -> ( ( w " { P } ) = ( v " { P } ) <-> ( w " { P } ) = ( w " { P } ) ) )
17	16	rspcev 3118	. . . . . . . . . 10 |- ( ( w e. U /\ ( w " { P } ) = ( w " { P } ) ) -> E. v e. U ( w " { P } ) = ( v " { P } ) )
18	14, 17	mpan2 682	. . . . . . . . 9 |- ( w e. U -> E. v e. U ( w " { P } ) = ( v " { P } ) )
19		vex 3016	. . . . . . . . . 10 |- w e. _V
20		imaexg 6718	. . . . . . . . . 10 |- ( w e. _V -> ( w " { P } ) e. _V )
21		eqid 2452	. . . . . . . . . . 11 |- ( v e. U |-> ( v " { P } ) ) = ( v e. U |-> ( v " { P } ) )
22	21	elrnmpt 5059	. . . . . . . . . 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 217	. . . . . . . 8 |- ( w e. U -> ( w " { P } ) e. ran ( v e. U |-> ( v " { P } ) ) )
25	24	ad2antlr 738	. . . . . . 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 21287	. . . . . . . . . . . 12 |- ( W e. UnifSp <-> ( U e. (UnifOn ` X ) /\ J = (unifTop ` U ) ) )
28	27	simplbi 466	. . . . . . . . . . 11 |- ( W e. UnifSp -> U e. (UnifOn ` X ) )
29	28	3ad2ant1 1030	. . . . . . . . . 10 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> U e. (UnifOn ` X ) )
30		eqid 2452	. . . . . . . . . . 11 |- (unifTop ` U ) = (unifTop ` U )
31	30	utopsnneip 21274	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( ( nei ` (unifTop ` U ) ) ` { P } ) = ran ( v e. U |-> ( v " { P } ) ) )
32	29, 6, 31	syl2anc 671	. . . . . . . . 9 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( nei ` (unifTop ` U ) ) ` { P } ) = ran ( v e. U |-> ( v " { P } ) ) )
33	32	eleq2d 2515	. . . . . . . 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 741	. . . . . . 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 240	. . . . . 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 1012	. . . . . . . . . 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 1235	. . . . . . . . 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 470	. . . . . . . . 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 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 2533	. . . . 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 467	. . . . 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 4838	. . . . . . 7 |- ( a = ( w " { P } ) -> ( a X. a ) = ( ( w " { P } ) X. ( w " { P } ) ) )
46	45	sseq1d 3427	. . . . . 6 |- ( a = ( w " { P } ) -> ( ( a X. a ) C_ v <-> ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) )
47	46	rspcev 3118	. . . . 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 671	. . . 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 471	. . . . 5 |- ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) -> U e. (UnifOn ` X ) )
50	6	adantr 471	. . . . 5 |- ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) -> P e. X )
51		simpr 467	. . . . 5 |- ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) -> v e. U )
52		simpll1 1048	. . . . . . . 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 767	. . . . . . . 8 |- ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) -> u e. U )
54		ustexsym 21241	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ u e. U ) -> E. w e. U ( `' w = w /\ w C_ u ) )
55	52, 53, 54	syl2anc 671	. . . . . . 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 737	. . . . . . . . . . . 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 767	. . . . . . . . . . . 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 21230	. . . . . . . . . . . 12 |- ( ( U e. (UnifOn ` X ) /\ w e. U ) -> w C_ ( X X. X ) )
59	56, 57, 58	syl2anc 671	. . . . . . . . . . 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 1049	. . . . . . . . . . . 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 1235	. . . . . . . . . . 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 21249	. . . . . . . . . . 11 |- ( ( w C_ ( X X. X ) /\ P e. X ) -> ( ( w " { P } ) X. ( w " { P } ) ) C_ ( w o. `' w ) )
63	59, 61, 62	syl2anc 671	. . . . . . . . . 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 769	. . . . . . . . . . . 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 4975	. . . . . . . . . . 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 4968	. . . . . . . . . . . . . 14 |- ( w C_ u -> ( w o. w ) C_ ( u o. w ) )
67		coss2 4969	. . . . . . . . . . . . . 14 |- ( w C_ u -> ( u o. w ) C_ ( u o. u ) )
68	66, 67	sstrd 3410	. . . . . . . . . . . . 13 |- ( w C_ u -> ( w o. w ) C_ ( u o. u ) )
69	68	ad2antll 740	. . . . . . . . . . . 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 774	. . . . . . . . . . . 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 3410	. . . . . . . . . . 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 3434	. . . . . . . . . 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 3410	. . . . . . . . 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 440	. . . . . . . 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 2839	. . . . . . 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 21236	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ v e. U ) -> E. u e. U ( u o. u ) C_ v )
78	77	3adant2 1028	. . . . . 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 2900	. . . . 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 1271	. . . 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 2900	. . 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 2790	. 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 21314	. . 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 933	1 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( nei ` J ) ` { P } ) e. (CauFilu ` U ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 986   = wceq 1448   e. wcel 1891   =/= wne 2622   A.wral 2737   E.wrex 2738   _Vcvv 3013   C_ wss 3372   (/)c0 3699   {csn 3936   |-> cmpt 4433   X. cxp 4810   `'ccnv 4811   ran crn 4813   "cima 4815   o. ccom 4816   ` cfv 5561   Basecbs 15132   TopOpenctopn 15331   fBascfbas 18969  TopOnctopon 19929   TopSpctps 19930   neicnei 20124   Filcfil 20871  UnifOncust 21225  unifTopcutop 21256  UnifStcuss 21279  UnifSpcusp 21280  CauFil_uccfilu 21312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-rep 4487  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4169  df-int 4205  df-iun 4250  df-br 4375  df-opab 4434  df-mpt 4435  df-tr 4470  df-eprel 4723  df-id 4727  df-po 4733  df-so 4734  df-fr 4771  df-we 4773  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-pred 5359  df-ord 5405  df-on 5406  df-lim 5407  df-suc 5408  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-om 6681  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-en 7557  df-fin 7560  df-fi 7912  df-fbas 18978  df-top 19932  df-topon 19934  df-topsp 19935  df-nei 20125  df-fil 20872  df-ust 21226  df-utop 21257  df-usp 21283  df-cfilu 21313

This theorem is referenced by:  ucnextcn  21330