Theorem blbas 22235

Description: The balls of a metric space form a basis for a topology. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 15-Jan-2014.)

Assertion

Ref	Expression
blbas	|- ( D e. ( *Met ` X ) -> ran ( ball ` D ) e. TopBases )

Proof of Theorem blbas

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

Step	Hyp	Ref	Expression
1		blin2 22234	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ z e. ( x i^i y ) ) /\ ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) ) -> E. r e. RR+ ( z ( ball ` D ) r ) C_ ( x i^i y ) )
2		simpll 790	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ z e. ( x i^i y ) ) /\ ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) ) -> D e. ( *Met ` X ) )
3		inss1 3833	. . . . . . . . . . 11 |- ( x i^i y ) C_ x
4	3	sseli 3599	. . . . . . . . . 10 |- ( z e. ( x i^i y ) -> z e. x )
5		elunii 4441	. . . . . . . . . 10 |- ( ( z e. x /\ x e. ran ( ball ` D ) ) -> z e. U. ran ( ball ` D ) )
6	4, 5	sylan 488	. . . . . . . . 9 |- ( ( z e. ( x i^i y ) /\ x e. ran ( ball ` D ) ) -> z e. U. ran ( ball ` D ) )
7	6	ad2ant2lr 784	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ z e. ( x i^i y ) ) /\ ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) ) -> z e. U. ran ( ball ` D ) )
8		unirnbl 22225	. . . . . . . . 9 |- ( D e. ( *Met ` X ) -> U. ran ( ball ` D ) = X )
9	8	ad2antrr 762	. . . . . . . 8 |- ( ( ( D e. ( *Met ` X ) /\ z e. ( x i^i y ) ) /\ ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) ) -> U. ran ( ball ` D ) = X )
10	7, 9	eleqtrd 2703	. . . . . . 7 |- ( ( ( D e. ( *Met ` X ) /\ z e. ( x i^i y ) ) /\ ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) ) -> z e. X )
11		blssex 22232	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ z e. X ) -> ( E. b e. ran ( ball ` D ) ( z e. b /\ b C_ ( x i^i y ) ) <-> E. r e. RR+ ( z ( ball ` D ) r ) C_ ( x i^i y ) ) )
12	2, 10, 11	syl2anc 693	. . . . . 6 |- ( ( ( D e. ( *Met ` X ) /\ z e. ( x i^i y ) ) /\ ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) ) -> ( E. b e. ran ( ball ` D ) ( z e. b /\ b C_ ( x i^i y ) ) <-> E. r e. RR+ ( z ( ball ` D ) r ) C_ ( x i^i y ) ) )
13	1, 12	mpbird 247	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ z e. ( x i^i y ) ) /\ ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) ) -> E. b e. ran ( ball ` D ) ( z e. b /\ b C_ ( x i^i y ) ) )
14	13	ex 450	. . . 4 |- ( ( D e. ( *Met ` X ) /\ z e. ( x i^i y ) ) -> ( ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) -> E. b e. ran ( ball ` D ) ( z e. b /\ b C_ ( x i^i y ) ) ) )
15	14	ralrimdva 2969	. . 3 |- ( D e. ( *Met ` X ) -> ( ( x e. ran ( ball ` D ) /\ y e. ran ( ball ` D ) ) -> A. z e. ( x i^i y ) E. b e. ran ( ball ` D ) ( z e. b /\ b C_ ( x i^i y ) ) ) )
16	15	ralrimivv 2970	. 2 |- ( D e. ( *Met ` X ) -> A. x e. ran ( ball ` D ) A. y e. ran ( ball ` D ) A. z e. ( x i^i y ) E. b e. ran ( ball ` D ) ( z e. b /\ b C_ ( x i^i y ) ) )
17		fvex 6201	. . . 4 |- ( ball ` D ) e. _V
18	17	rnex 7100	. . 3 |- ran ( ball ` D ) e. _V
19		isbasis2g 20752	. . 3 |- ( ran ( ball ` D ) e. _V -> ( ran ( ball ` D ) e. TopBases <-> A. x e. ran ( ball ` D ) A. y e. ran ( ball ` D ) A. z e. ( x i^i y ) E. b e. ran ( ball ` D ) ( z e. b /\ b C_ ( x i^i y ) ) ) )
20	18, 19	ax-mp 5	. 2 |- ( ran ( ball ` D ) e. TopBases <-> A. x e. ran ( ball ` D ) A. y e. ran ( ball ` D ) A. z e. ( x i^i y ) E. b e. ran ( ball ` D ) ( z e. b /\ b C_ ( x i^i y ) ) )
21	16, 20	sylibr 224	1 |- ( D e. ( *Met ` X ) -> ran ( ball ` D ) e. TopBases )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   U.cuni 4436   ran crn 5115   ` cfv 5888  (class class class)co 6650   RR+crp 11832   *Metcxmt 19731   ballcbl 19733   TopBasesctb 20749

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-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-iun 4522  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-psmet 19738  df-xmet 19739  df-bl 19741  df-bases 20750

This theorem is referenced by:  mopntopon  22244  elmopn  22247  imasf1oxms  22294  blssopn  22300  metss  22313