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Theorem blbas 20140
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 20139 . . . . . 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 3681 . . . . . . . . . . 11  |-  ( x  i^i  y )  C_  x
43sseli 3463 . . . . . . . . . 10  |-  ( z  e.  ( x  i^i  y )  ->  z  e.  x )
5 elunii 4207 . . . . . . . . . 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 20130 . . . . . . . . 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 2544 . . . . . . 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 20137 . . . . . . 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 2912 . . 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 2913 . 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 5812 . . . 4  |-  ( ball `  D )  e.  _V
1817rnex 6625 . . 3  |-  ran  ( ball `  D )  e. 
_V
19 isbasis2g 18688 . . 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 1370    e. wcel 1758   A.wral 2799   E.wrex 2800   _Vcvv 3078    i^i cin 3438    C_ wss 3439   U.cuni 4202   ran crn 4952   ` cfv 5529  (class class class)co 6203   RR+crp 11105   *Metcxmt 17929   ballcbl 17931   TopBasesctb 18637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-q 11068  df-rp 11106  df-xneg 11203  df-xadd 11204  df-xmul 11205  df-psmet 17937  df-xmet 17938  df-bl 17940  df-bases 18640
This theorem is referenced by:  mopntopon  20149  elmopn  20152  imasf1oxms  20199  blssopn  20205  metss  20218
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