Theorem neipcfilu 19876

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 989	. . . . 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 18546	. . . . 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 990	. . . . 5 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> P e. X )
7	6	snssd 4023	. . . 4 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> { P } C_ X )
8		snnzg 3997	. . . . 5 |- ( P e. X -> { P } =/= (/) )
9	6, 8	syl 16	. . . 4 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> { P } =/= (/) )
10		neifil 19458	. . . 4 |- ( ( J e. (TopOn ` X ) /\ { P } C_ X /\ { P } =/= (/) ) -> ( ( nei ` J ) ` { P } ) e. ( Fil ` X ) )
11	5, 7, 9, 10	syl3anc 1218	. . 3 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( nei ` J ) ` { P } ) e. ( Fil ` X ) )
12		filfbas 19426	. . 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 2443	. . . . . . . . . 10 |- ( w " { P } ) = ( w " { P } )
15		imaeq1 5169	. . . . . . . . . . . 12 |- ( v = w -> ( v " { P } ) = ( w " { P } ) )
16	15	eqeq2d 2454	. . . . . . . . . . 11 |- ( v = w -> ( ( w " { P } ) = ( v " { P } ) <-> ( w " { P } ) = ( w " { P } ) ) )
17	16	rspcev 3078	. . . . . . . . . 10 |- ( ( w e. U /\ ( w " { P } ) = ( w " { P } ) ) -> E. v e. U ( w " { P } ) = ( v " { P } ) )
18	14, 17	mpan2 671	. . . . . . . . 9 |- ( w e. U -> E. v e. U ( w " { P } ) = ( v " { P } ) )
19		vex 2980	. . . . . . . . . 10 |- w e. _V
20		imaexg 6520	. . . . . . . . . 10 |- ( w e. _V -> ( w " { P } ) e. _V )
21		eqid 2443	. . . . . . . . . . 11 |- ( v e. U |-> ( v " { P } ) ) = ( v e. U |-> ( v " { P } ) )
22	21	elrnmpt 5091	. . . . . . . . . 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 726	. . . . . . 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 19841	. . . . . . . . . . . 12 |- ( W e. UnifSp <-> ( U e. (UnifOn ` X ) /\ J = (unifTop ` U ) ) )
28	27	simplbi 460	. . . . . . . . . . 11 |- ( W e. UnifSp -> U e. (UnifOn ` X ) )
29	28	3ad2ant1 1009	. . . . . . . . . 10 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> U e. (UnifOn ` X ) )
30		eqid 2443	. . . . . . . . . . 11 |- (unifTop ` U ) = (unifTop ` U )
31	30	utopsnneip 19828	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ P e. X ) -> ( ( nei ` (unifTop ` U ) ) ` { P } ) = ran ( v e. U |-> ( v " { P } ) ) )
32	29, 6, 31	syl2anc 661	. . . . . . . . 9 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( nei ` (unifTop ` U ) ) ` { P } ) = ran ( v e. U |-> ( v " { P } ) ) )
33	32	eleq2d 2510	. . . . . . . 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 729	. . . . . . 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 991	. . . . . . . . . 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 1209	. . . . . . . . 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 464	. . . . . . . . 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 5700	. . . . . . 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 5698	. . . . . 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 2520	. . . . 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 461	. . . . 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, 44	xpeq12d 4870	. . . . . . 7 |- ( a = ( w " { P } ) -> ( a X. a ) = ( ( w " { P } ) X. ( w " { P } ) ) )
46	45	sseq1d 3388	. . . . . 6 |- ( a = ( w " { P } ) -> ( ( a X. a ) C_ v <-> ( ( w " { P } ) X. ( w " { P } ) ) C_ v ) )
47	46	rspcev 3078	. . . . 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 661	. . . 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 465	. . . . 5 |- ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) -> U e. (UnifOn ` X ) )
50	6	adantr 465	. . . . 5 |- ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) -> P e. X )
51		simpr 461	. . . . 5 |- ( ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) /\ v e. U ) -> v e. U )
52		simpll1 1027	. . . . . . . 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 754	. . . . . . . 8 |- ( ( ( ( U e. (UnifOn ` X ) /\ P e. X /\ v e. U ) /\ u e. U ) /\ ( u o. u ) C_ v ) -> u e. U )
54		ustexsym 19795	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ u e. U ) -> E. w e. U ( `' w = w /\ w C_ u ) )
55	52, 53, 54	syl2anc 661	. . . . . . 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 725	. . . . . . . . . . . 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 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 e. U )
58		ustssxp 19784	. . . . . . . . . . . 12 |- ( ( U e. (UnifOn ` X ) /\ w e. U ) -> w C_ ( X X. X ) )
59	56, 57, 58	syl2anc 661	. . . . . . . . . . 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 1028	. . . . . . . . . . . 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 1209	. . . . . . . . . . 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 19803	. . . . . . . . . . 11 |- ( ( w C_ ( X X. X ) /\ P e. X ) -> ( ( w " { P } ) X. ( w " { P } ) ) C_ ( w o. `' w ) )
63	59, 61, 62	syl2anc 661	. . . . . . . . . 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 755	. . . . . . . . . . . 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 5007	. . . . . . . . . . 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 5000	. . . . . . . . . . . . . 14 |- ( w C_ u -> ( w o. w ) C_ ( u o. w ) )
67		coss2 5001	. . . . . . . . . . . . . 14 |- ( w C_ u -> ( u o. w ) C_ ( u o. u ) )
68	66, 67	sstrd 3371	. . . . . . . . . . . . 13 |- ( w C_ u -> ( w o. w ) C_ ( u o. u ) )
69	68	ad2antll 728	. . . . . . . . . . . 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 3371	. . . . . . . . . . 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 3395	. . . . . . . . . 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 3371	. . . . . . . . 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 434	. . . . . . . 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 2833	. . . . . . 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 19790	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ v e. U ) -> E. u e. U ( u o. u ) C_ v )
78	77	3adant2 1007	. . . . . 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 2867	. . . . 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 1218	. . . 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 2867	. . 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 2804	. 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 19868	. . 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 913	1 |- ( ( W e. UnifSp /\ W e. TopSp /\ P e. X ) -> ( ( nei ` J ) ` { P } ) e. (CauFilu ` U ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2611   A.wral 2720   E.wrex 2721   _Vcvv 2977   C_ wss 3333   (/)c0 3642   {csn 3882   e. cmpt 4355   X. cxp 4843   `'ccnv 4844   ran crn 4846   "cima 4848   o. ccom 4849   ` cfv 5423   Basecbs 14179   TopOpenctopn 14365   fBascfbas 17809  TopOnctopon 18504   TopSpctps 18506   neicnei 18706   Filcfil 19423  UnifOncust 19779  unifTopcutop 19810  UnifStcuss 19833  UnifSpcusp 19834  CauFil_uccfilu 19866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-fin 7319  df-fi 7666  df-fbas 17819  df-top 18508  df-topon 18511  df-topsp 18512  df-nei 18707  df-fil 19424  df-ust 19780  df-utop 19811  df-usp 19837  df-cfilu 19867

This theorem is referenced by:  ucnextcn  19884