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Theorem blbas 17992
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 17991 . . . . . 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 730 . . . . . . 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 3402 . . . . . . . . . . 11  |-  ( x  i^i  y )  C_  x
43sseli 3189 . . . . . . . . . 10  |-  ( z  e.  ( x  i^i  y )  ->  z  e.  x )
5 elunii 3848 . . . . . . . . . 10  |-  ( ( z  e.  x  /\  x  e.  ran  ( ball `  D ) )  -> 
z  e.  U. ran  ( ball `  D )
)
64, 5sylan 457 . . . . . . . . 9  |-  ( ( z  e.  ( x  i^i  y )  /\  x  e.  ran  ( ball `  D ) )  -> 
z  e.  U. ran  ( ball `  D )
)
76ad2ant2lr 728 . . . . . . . 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 17985 . . . . . . . . 9  |-  ( D  e.  ( * Met `  X )  ->  U. ran  ( ball `  D )  =  X )
98ad2antrr 706 . . . . . . . 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 2372 . . . . . . 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 17989 . . . . . . 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 642 . . . . . 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 223 . . . . 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 423 . . . 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 2646 . . 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 2647 . 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 5555 . . . 4  |-  ( ball `  D )  e.  _V
1817rnex 4958 . . 3  |-  ran  ( ball `  D )  e. 
_V
19 isbasis2g 16702 . . 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 8 . 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 203 1  |-  ( D  e.  ( * Met `  X )  ->  ran  ( ball `  D )  e. 
TopBases )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   U.cuni 3843   ran crn 4706   ` cfv 5271  (class class class)co 5874   RR+crp 10370   * Metcxmt 16385   ballcbl 16387   TopBasesctb 16651
This theorem is referenced by:  mopntopon  18001  elmopn  18004  imasf1oxms  18051  blssopn  18057  metss  18070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-xmet 16389  df-bl 16391  df-bases 16654
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