Theorem neifil 21684

Description: The neighborhoods of a nonempty set is a filter. Example 2 of [BourbakiTop1] p. I.36. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
neifil	|- ( ( J e. (TopOn ` X ) /\ S C_ X /\ S =/= (/) ) -> ( ( nei ` J ) ` S ) e. ( Fil ` X ) )

Proof of Theorem neifil

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		toponuni 20719	. . . . . . . 8 |- ( J e. (TopOn ` X ) -> X = U. J )
2	1	adantr 481	. . . . . . 7 |- ( ( J e. (TopOn ` X ) /\ S C_ X ) -> X = U. J )
3		topontop 20718	. . . . . . . . 9 |- ( J e. (TopOn ` X ) -> J e. Top )
4	3	adantr 481	. . . . . . . 8 |- ( ( J e. (TopOn ` X ) /\ S C_ X ) -> J e. Top )
5		simpr 477	. . . . . . . . 9 |- ( ( J e. (TopOn ` X ) /\ S C_ X ) -> S C_ X )
6	5, 2	sseqtrd 3641	. . . . . . . 8 |- ( ( J e. (TopOn ` X ) /\ S C_ X ) -> S C_ U. J )
7		eqid 2622	. . . . . . . . 9 |- U. J = U. J
8	7	neiuni 20926	. . . . . . . 8 |- ( ( J e. Top /\ S C_ U. J ) -> U. J = U. ( ( nei ` J ) ` S ) )
9	4, 6, 8	syl2anc 693	. . . . . . 7 |- ( ( J e. (TopOn ` X ) /\ S C_ X ) -> U. J = U. ( ( nei ` J ) ` S ) )
10	2, 9	eqtrd 2656	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ S C_ X ) -> X = U. ( ( nei ` J ) ` S ) )
11		eqimss2 3658	. . . . . 6 |- ( X = U. ( ( nei ` J ) ` S ) -> U. ( ( nei ` J ) ` S ) C_ X )
12	10, 11	syl 17	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ S C_ X ) -> U. ( ( nei ` J ) ` S ) C_ X )
13		sspwuni 4611	. . . . 5 |- ( ( ( nei ` J ) ` S ) C_ ~P X <-> U. ( ( nei ` J ) ` S ) C_ X )
14	12, 13	sylibr 224	. . . 4 |- ( ( J e. (TopOn ` X ) /\ S C_ X ) -> ( ( nei ` J ) ` S ) C_ ~P X )
15	14	3adant3 1081	. . 3 |- ( ( J e. (TopOn ` X ) /\ S C_ X /\ S =/= (/) ) -> ( ( nei ` J ) ` S ) C_ ~P X )
16		0nnei 20916	. . . . 5 |- ( ( J e. Top /\ S =/= (/) ) -> -. (/) e. ( ( nei ` J ) ` S ) )
17	3, 16	sylan 488	. . . 4 |- ( ( J e. (TopOn ` X ) /\ S =/= (/) ) -> -. (/) e. ( ( nei ` J ) ` S ) )
18	17	3adant2 1080	. . 3 |- ( ( J e. (TopOn ` X ) /\ S C_ X /\ S =/= (/) ) -> -. (/) e. ( ( nei ` J ) ` S ) )
19	7	tpnei 20925	. . . . . . 7 |- ( J e. Top -> ( S C_ U. J <-> U. J e. ( ( nei ` J ) ` S ) ) )
20	19	biimpa 501	. . . . . 6 |- ( ( J e. Top /\ S C_ U. J ) -> U. J e. ( ( nei ` J ) ` S ) )
21	4, 6, 20	syl2anc 693	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ S C_ X ) -> U. J e. ( ( nei ` J ) ` S ) )
22	2, 21	eqeltrd 2701	. . . 4 |- ( ( J e. (TopOn ` X ) /\ S C_ X ) -> X e. ( ( nei ` J ) ` S ) )
23	22	3adant3 1081	. . 3 |- ( ( J e. (TopOn ` X ) /\ S C_ X /\ S =/= (/) ) -> X e. ( ( nei ` J ) ` S ) )
24	15, 18, 23	3jca 1242	. 2 |- ( ( J e. (TopOn ` X ) /\ S C_ X /\ S =/= (/) ) -> ( ( ( nei ` J ) ` S ) C_ ~P X /\ -. (/) e. ( ( nei ` J ) ` S ) /\ X e. ( ( nei ` J ) ` S ) ) )
25		elpwi 4168	. . . . 5 |- ( x e. ~P X -> x C_ X )
26	4	ad2antrr 762	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ x C_ X ) /\ ( y e. ( ( nei ` J ) ` S ) /\ y C_ x ) ) -> J e. Top )
27		simprl 794	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ x C_ X ) /\ ( y e. ( ( nei ` J ) ` S ) /\ y C_ x ) ) -> y e. ( ( nei ` J ) ` S ) )
28		simprr 796	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ x C_ X ) /\ ( y e. ( ( nei ` J ) ` S ) /\ y C_ x ) ) -> y C_ x )
29		simplr 792	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ x C_ X ) /\ ( y e. ( ( nei ` J ) ` S ) /\ y C_ x ) ) -> x C_ X )
30	2	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ x C_ X ) /\ ( y e. ( ( nei ` J ) ` S ) /\ y C_ x ) ) -> X = U. J )
31	29, 30	sseqtrd 3641	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ x C_ X ) /\ ( y e. ( ( nei ` J ) ` S ) /\ y C_ x ) ) -> x C_ U. J )
32	7	ssnei2 20920	. . . . . . 7 |- ( ( ( J e. Top /\ y e. ( ( nei ` J ) ` S ) ) /\ ( y C_ x /\ x C_ U. J ) ) -> x e. ( ( nei ` J ) ` S ) )
33	26, 27, 28, 31, 32	syl22anc 1327	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ x C_ X ) /\ ( y e. ( ( nei ` J ) ` S ) /\ y C_ x ) ) -> x e. ( ( nei ` J ) ` S ) )
34	33	rexlimdvaa 3032	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ x C_ X ) -> ( E. y e. ( ( nei ` J ) ` S ) y C_ x -> x e. ( ( nei ` J ) ` S ) ) )
35	25, 34	sylan2 491	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ x e. ~P X ) -> ( E. y e. ( ( nei ` J ) ` S ) y C_ x -> x e. ( ( nei ` J ) ` S ) ) )
36	35	ralrimiva 2966	. . 3 |- ( ( J e. (TopOn ` X ) /\ S C_ X ) -> A. x e. ~P X ( E. y e. ( ( nei ` J ) ` S ) y C_ x -> x e. ( ( nei ` J ) ` S ) ) )
37	36	3adant3 1081	. 2 |- ( ( J e. (TopOn ` X ) /\ S C_ X /\ S =/= (/) ) -> A. x e. ~P X ( E. y e. ( ( nei ` J ) ` S ) y C_ x -> x e. ( ( nei ` J ) ` S ) ) )
38		innei 20929	. . . . . 6 |- ( ( J e. Top /\ x e. ( ( nei ` J ) ` S ) /\ y e. ( ( nei ` J ) ` S ) ) -> ( x i^i y ) e. ( ( nei ` J ) ` S ) )
39	38	3expib 1268	. . . . 5 |- ( J e. Top -> ( ( x e. ( ( nei ` J ) ` S ) /\ y e. ( ( nei ` J ) ` S ) ) -> ( x i^i y ) e. ( ( nei ` J ) ` S ) ) )
40	3, 39	syl 17	. . . 4 |- ( J e. (TopOn ` X ) -> ( ( x e. ( ( nei ` J ) ` S ) /\ y e. ( ( nei ` J ) ` S ) ) -> ( x i^i y ) e. ( ( nei ` J ) ` S ) ) )
41	40	3ad2ant1 1082	. . 3 |- ( ( J e. (TopOn ` X ) /\ S C_ X /\ S =/= (/) ) -> ( ( x e. ( ( nei ` J ) ` S ) /\ y e. ( ( nei ` J ) ` S ) ) -> ( x i^i y ) e. ( ( nei ` J ) ` S ) ) )
42	41	ralrimivv 2970	. 2 |- ( ( J e. (TopOn ` X ) /\ S C_ X /\ S =/= (/) ) -> A. x e. ( ( nei ` J ) ` S ) A. y e. ( ( nei ` J ) ` S ) ( x i^i y ) e. ( ( nei ` J ) ` S ) )
43		isfil2 21660	. 2 |- ( ( ( nei ` J ) ` S ) e. ( Fil ` X ) <-> ( ( ( ( nei ` J ) ` S ) C_ ~P X /\ -. (/) e. ( ( nei ` J ) ` S ) /\ X e. ( ( nei ` J ) ` S ) ) /\ A. x e. ~P X ( E. y e. ( ( nei ` J ) ` S ) y C_ x -> x e. ( ( nei ` J ) ` S ) ) /\ A. x e. ( ( nei ` J ) ` S ) A. y e. ( ( nei ` J ) ` S ) ( x i^i y ) e. ( ( nei ` J ) ` S ) ) )
44	24, 37, 42, 43	syl3anbrc 1246	1 |- ( ( J e. (TopOn ` X ) /\ S C_ X /\ S =/= (/) ) -> ( ( nei ` J ) ` S ) e. ( Fil ` X ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   ` cfv 5888   Topctop 20698  TopOnctopon 20715   neicnei 20901   Filcfil 21649

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-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-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-fbas 19743  df-top 20699  df-topon 20716  df-nei 20902  df-fil 21650

This theorem is referenced by:  trnei  21696  neiflim  21778  hausflim  21785  flimcf  21786  flimclslem  21788  cnpflf2  21804  cnpflf  21805  fclsfnflim  21831  neipcfilu  22100