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Theorem blbas 19980
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.
StepHypRef Expression
1 blin2 19979 . . . . . 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 753 . . . . . . 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 3565 . . . . . . . . . . 11  |-  ( x  i^i  y )  C_  x
43sseli 3347 . . . . . . . . . 10  |-  ( z  e.  ( x  i^i  y )  ->  z  e.  x )
5 elunii 4091 . . . . . . . . . 10  |-  ( ( z  e.  x  /\  x  e.  ran  ( ball `  D ) )  -> 
z  e.  U. ran  ( ball `  D )
)
64, 5sylan 471 . . . . . . . . 9  |-  ( ( z  e.  ( x  i^i  y )  /\  x  e.  ran  ( ball `  D ) )  -> 
z  e.  U. ran  ( ball `  D )
)
76ad2ant2lr 747 . . . . . . . 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 19970 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  U. ran  ( ball `  D )  =  X )
98ad2antrr 725 . . . . . . . 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 )
107, 9eleqtrd 2514 . . . . . . 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 19977 . . . . . . 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 )
) )
122, 10, 11syl2anc 661 . . . . . 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 )
) )
131, 12mpbird 232 . . . . 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 )
) )
1413ex 434 . . . 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 )
) ) )
1514ralrimdva 2801 . . 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
) ) ) )
1615ralrimivv 2802 . 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 5696 . . . 4  |-  ( ball `  D )  e.  _V
1817rnex 6507 . . 3  |-  ran  ( ball `  D )  e. 
_V
19 isbasis2g 18528 . . 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
) ) ) )
2018, 19ax-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
) ) )
2116, 20sylibr 212 1  |-  ( D  e.  ( *Met `  X )  ->  ran  ( ball `  D )  e. 
TopBases )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710   E.wrex 2711   _Vcvv 2967    i^i cin 3322    C_ wss 3323   U.cuni 4086   ran crn 4836   ` cfv 5413  (class class class)co 6086   RR+crp 10983   *Metcxmt 17776   ballcbl 17778   TopBasesctb 18477
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-psmet 17784  df-xmet 17785  df-bl 17787  df-bases 18480
This theorem is referenced by:  mopntopon  19989  elmopn  19992  imasf1oxms  20039  blssopn  20045  metss  20058
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