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Theorem 0totbnd 26497
Description: The metric (there is only one) on the empty set is totally bounded. (Contributed by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
0totbnd  |-  ( X  =  (/)  ->  ( M  e.  ( TotBnd `  X
)  <->  M  e.  ( Met `  X ) ) )

Proof of Theorem 0totbnd
Dummy variables  v 
r  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . 3  |-  ( X  =  (/)  ->  ( TotBnd `  X )  =  (
TotBnd `  (/) ) )
21eleq2d 2350 . 2  |-  ( X  =  (/)  ->  ( M  e.  ( TotBnd `  X
)  <->  M  e.  ( TotBnd `
 (/) ) ) )
3 fveq2 5525 . . . 4  |-  ( X  =  (/)  ->  ( Met `  X )  =  ( Met `  (/) ) )
43eleq2d 2350 . . 3  |-  ( X  =  (/)  ->  ( M  e.  ( Met `  X
)  <->  M  e.  ( Met `  (/) ) ) )
5 0elpw 4180 . . . . . . 7  |-  (/)  e.  ~P (/)
6 0fin 7087 . . . . . . 7  |-  (/)  e.  Fin
7 elin 3358 . . . . . . 7  |-  ( (/)  e.  ( ~P (/)  i^i  Fin ) 
<->  ( (/)  e.  ~P (/) 
/\  (/)  e.  Fin )
)
85, 6, 7mpbir2an 886 . . . . . 6  |-  (/)  e.  ( ~P (/)  i^i  Fin )
9 0iun 3959 . . . . . 6  |-  U_ x  e.  (/)  ( x (
ball `  M )
r )  =  (/)
10 iuneq1 3918 . . . . . . . 8  |-  ( v  =  (/)  ->  U_ x  e.  v  ( x
( ball `  M )
r )  =  U_ x  e.  (/)  ( x ( ball `  M
) r ) )
1110eqeq1d 2291 . . . . . . 7  |-  ( v  =  (/)  ->  ( U_ x  e.  v  (
x ( ball `  M
) r )  =  (/) 
<-> 
U_ x  e.  (/)  ( x ( ball `  M ) r )  =  (/) ) )
1211rspcev 2884 . . . . . 6  |-  ( (
(/)  e.  ( ~P (/) 
i^i  Fin )  /\  U_ x  e.  (/)  ( x ( ball `  M
) r )  =  (/) )  ->  E. v  e.  ( ~P (/)  i^i  Fin ) U_ x  e.  v  ( x ( ball `  M ) r )  =  (/) )
138, 9, 12mp2an 653 . . . . 5  |-  E. v  e.  ( ~P (/)  i^i  Fin ) U_ x  e.  v  ( x ( ball `  M ) r )  =  (/)
1413rgenw 2610 . . . 4  |-  A. r  e.  RR+  E. v  e.  ( ~P (/)  i^i  Fin ) U_ x  e.  v  ( x ( ball `  M ) r )  =  (/)
15 istotbnd3 26495 . . . 4  |-  ( M  e.  ( TotBnd `  (/) )  <->  ( M  e.  ( Met `  (/) )  /\  A. r  e.  RR+  E. v  e.  ( ~P (/)  i^i  Fin ) U_ x  e.  v  ( x ( ball `  M ) r )  =  (/) ) )
1614, 15mpbiran2 885 . . 3  |-  ( M  e.  ( TotBnd `  (/) )  <->  M  e.  ( Met `  (/) ) )
174, 16syl6rbbr 255 . 2  |-  ( X  =  (/)  ->  ( M  e.  ( TotBnd `  (/) )  <->  M  e.  ( Met `  X ) ) )
182, 17bitrd 244 1  |-  ( X  =  (/)  ->  ( M  e.  ( TotBnd `  X
)  <->  M  e.  ( Met `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151   (/)c0 3455   ~Pcpw 3625   U_ciun 3905   ` cfv 5255  (class class class)co 5858   Fincfn 6863   RR+crp 10354   Metcme 16370   ballcbl 16371   TotBndctotbnd 26490
This theorem is referenced by:  prdsbnd2  26519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-fin 6867  df-totbnd 26492
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