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Theorem blbas 18462
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 18461 . . . . . 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 732 . . . . . . 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 3563 . . . . . . . . . . 11  |-  ( x  i^i  y )  C_  x
43sseli 3346 . . . . . . . . . 10  |-  ( z  e.  ( x  i^i  y )  ->  z  e.  x )
5 elunii 4022 . . . . . . . . . 10  |-  ( ( z  e.  x  /\  x  e.  ran  ( ball `  D ) )  -> 
z  e.  U. ran  ( ball `  D )
)
64, 5sylan 459 . . . . . . . . 9  |-  ( ( z  e.  ( x  i^i  y )  /\  x  e.  ran  ( ball `  D ) )  -> 
z  e.  U. ran  ( ball `  D )
)
76ad2ant2lr 730 . . . . . . . 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 18452 . . . . . . . . 9  |-  ( D  e.  ( * Met `  X )  ->  U. ran  ( ball `  D )  =  X )
98ad2antrr 708 . . . . . . . 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 18459 . . . . . . 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 644 . . . . . 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 225 . . . . 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 425 . . . 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 2798 . . 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 2799 . 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 5744 . . . 4  |-  ( ball `  D )  e.  _V
1817rnex 5135 . . 3  |-  ran  ( ball `  D )  e. 
_V
19 isbasis2g 17015 . . 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 205 1  |-  ( D  e.  ( * Met `  X )  ->  ran  ( ball `  D )  e. 
TopBases )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958    i^i cin 3321    C_ wss 3322   U.cuni 4017   ran crn 4881   ` cfv 5456  (class class class)co 6083   RR+crp 10614   * Metcxmt 16688   ballcbl 16690   TopBasesctb 16964
This theorem is referenced by:  mopntopon  18471  elmopn  18474  imasf1oxms  18521  blssopn  18527  metss  18540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-psmet 16696  df-xmet 16697  df-bl 16699  df-bases 16967
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