Theorem lpbl 22308

Description: Every ball around a limit point P of a subset S includes a member of S (even if P e/ S). (Contributed by NM, 9-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)

Hypothesis

Ref	Expression
mopni.1	|- J = ( MetOpen ` D )

Assertion

Ref	Expression
lpbl	|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ P e. ( ( limPt ` J ) ` S ) ) /\ R e. RR+ ) -> E. x e. S x e. ( P ( ball ` D ) R ) )

Distinct variable groups:   x, D   x, J   x, R   x, S   x, P   x, X

Proof of Theorem lpbl

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ P e. ( ( limPt ` J ) ` S ) ) /\ R e. RR+ ) -> D e. ( *Met ` X ) )
2		mopni.1	. . . . . . . . 9 |- J = ( MetOpen ` D )
3	2	mopntop 22245	. . . . . . . 8 |- ( D e. ( *Met ` X ) -> J e. Top )
4	1, 3	syl 17	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ P e. ( ( limPt ` J ) ` S ) ) /\ R e. RR+ ) -> J e. Top )
5		simpl2 1065	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ P e. ( ( limPt ` J ) ` S ) ) /\ R e. RR+ ) -> S C_ X )
6	2	mopnuni 22246	. . . . . . . . 9 |- ( D e. ( *Met ` X ) -> X = U. J )
7	1, 6	syl 17	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ P e. ( ( limPt ` J ) ` S ) ) /\ R e. RR+ ) -> X = U. J )
8	5, 7	sseqtrd 3641	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ P e. ( ( limPt ` J ) ` S ) ) /\ R e. RR+ ) -> S C_ U. J )
9		eqid 2622	. . . . . . . 8 |- U. J = U. J
10	9	lpss 20946	. . . . . . 7 |- ( ( J e. Top /\ S C_ U. J ) -> ( ( limPt ` J ) ` S ) C_ U. J )
11	4, 8, 10	syl2anc 693	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ P e. ( ( limPt ` J ) ` S ) ) /\ R e. RR+ ) -> ( ( limPt ` J ) ` S ) C_ U. J )
12		simpl3 1066	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ P e. ( ( limPt ` J ) ` S ) ) /\ R e. RR+ ) -> P e. ( ( limPt ` J ) ` S ) )
13	11, 12	sseldd 3604	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ P e. ( ( limPt ` J ) ` S ) ) /\ R e. RR+ ) -> P e. U. J )
14	13, 7	eleqtrrd 2704	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ P e. ( ( limPt ` J ) ` S ) ) /\ R e. RR+ ) -> P e. X )
15		simpr 477	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ P e. ( ( limPt ` J ) ` S ) ) /\ R e. RR+ ) -> R e. RR+ )
16	2	blnei 22307	. . . 4 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ R e. RR+ ) -> ( P ( ball ` D ) R ) e. ( ( nei ` J ) ` { P } ) )
17	1, 14, 15, 16	syl3anc 1326	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ P e. ( ( limPt ` J ) ` S ) ) /\ R e. RR+ ) -> ( P ( ball ` D ) R ) e. ( ( nei ` J ) ` { P } ) )
18	9	islp2 20949	. . . . 5 |- ( ( J e. Top /\ S C_ U. J /\ P e. U. J ) -> ( P e. ( ( limPt ` J ) ` S ) <-> A. x e. ( ( nei ` J ) ` { P } ) ( x i^i ( S \ { P } ) ) =/= (/) ) )
19	4, 8, 13, 18	syl3anc 1326	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ P e. ( ( limPt ` J ) ` S ) ) /\ R e. RR+ ) -> ( P e. ( ( limPt ` J ) ` S ) <-> A. x e. ( ( nei ` J ) ` { P } ) ( x i^i ( S \ { P } ) ) =/= (/) ) )
20	12, 19	mpbid 222	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ P e. ( ( limPt ` J ) ` S ) ) /\ R e. RR+ ) -> A. x e. ( ( nei ` J ) ` { P } ) ( x i^i ( S \ { P } ) ) =/= (/) )
21		ineq1 3807	. . . . 5 |- ( x = ( P ( ball ` D ) R ) -> ( x i^i ( S \ { P } ) ) = ( ( P ( ball ` D ) R ) i^i ( S \ { P } ) ) )
22	21	neeq1d 2853	. . . 4 |- ( x = ( P ( ball ` D ) R ) -> ( ( x i^i ( S \ { P } ) ) =/= (/) <-> ( ( P ( ball ` D ) R ) i^i ( S \ { P } ) ) =/= (/) ) )
23	22	rspcva 3307	. . 3 |- ( ( ( P ( ball ` D ) R ) e. ( ( nei ` J ) ` { P } ) /\ A. x e. ( ( nei ` J ) ` { P } ) ( x i^i ( S \ { P } ) ) =/= (/) ) -> ( ( P ( ball ` D ) R ) i^i ( S \ { P } ) ) =/= (/) )
24	17, 20, 23	syl2anc 693	. 2 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ P e. ( ( limPt ` J ) ` S ) ) /\ R e. RR+ ) -> ( ( P ( ball ` D ) R ) i^i ( S \ { P } ) ) =/= (/) )
25		elin 3796	. . . . 5 |- ( x e. ( ( P ( ball ` D ) R ) i^i ( S \ { P } ) ) <-> ( x e. ( P ( ball ` D ) R ) /\ x e. ( S \ { P } ) ) )
26		eldifi 3732	. . . . . . 7 |- ( x e. ( S \ { P } ) -> x e. S )
27	26	anim2i 593	. . . . . 6 |- ( ( x e. ( P ( ball ` D ) R ) /\ x e. ( S \ { P } ) ) -> ( x e. ( P ( ball ` D ) R ) /\ x e. S ) )
28	27	ancomd 467	. . . . 5 |- ( ( x e. ( P ( ball ` D ) R ) /\ x e. ( S \ { P } ) ) -> ( x e. S /\ x e. ( P ( ball ` D ) R ) ) )
29	25, 28	sylbi 207	. . . 4 |- ( x e. ( ( P ( ball ` D ) R ) i^i ( S \ { P } ) ) -> ( x e. S /\ x e. ( P ( ball ` D ) R ) ) )
30	29	eximi 1762	. . 3 |- ( E. x x e. ( ( P ( ball ` D ) R ) i^i ( S \ { P } ) ) -> E. x ( x e. S /\ x e. ( P ( ball ` D ) R ) ) )
31		n0 3931	. . 3 |- ( ( ( P ( ball ` D ) R ) i^i ( S \ { P } ) ) =/= (/) <-> E. x x e. ( ( P ( ball ` D ) R ) i^i ( S \ { P } ) ) )
32		df-rex 2918	. . 3 |- ( E. x e. S x e. ( P ( ball ` D ) R ) <-> E. x ( x e. S /\ x e. ( P ( ball ` D ) R ) ) )
33	30, 31, 32	3imtr4i 281	. 2 |- ( ( ( P ( ball ` D ) R ) i^i ( S \ { P } ) ) =/= (/) -> E. x e. S x e. ( P ( ball ` D ) R ) )
34	24, 33	syl 17	1 |- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ P e. ( ( limPt ` J ) ` S ) ) /\ R e. RR+ ) -> E. x e. S x e. ( P ( ball ` D ) R ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   U.cuni 4436   ` cfv 5888  (class class class)co 6650   RR+crp 11832   *Metcxmt 19731   ballcbl 19733   MetOpencmopn 19736   Topctop 20698   neicnei 20901   limPtclp 20938

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

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-rmo 2920  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-iin 4523  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940

This theorem is referenced by:  limcrecl  39861