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Theorem 0totbnd 26600
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 5541 . . 3  |-  ( X  =  (/)  ->  ( TotBnd `  X )  =  (
TotBnd `  (/) ) )
21eleq2d 2363 . 2  |-  ( X  =  (/)  ->  ( M  e.  ( TotBnd `  X
)  <->  M  e.  ( TotBnd `
 (/) ) ) )
3 fveq2 5541 . . . 4  |-  ( X  =  (/)  ->  ( Met `  X )  =  ( Met `  (/) ) )
43eleq2d 2363 . . 3  |-  ( X  =  (/)  ->  ( M  e.  ( Met `  X
)  <->  M  e.  ( Met `  (/) ) ) )
5 0elpw 4196 . . . . . . 7  |-  (/)  e.  ~P (/)
6 0fin 7103 . . . . . . 7  |-  (/)  e.  Fin
7 elin 3371 . . . . . . 7  |-  ( (/)  e.  ( ~P (/)  i^i  Fin ) 
<->  ( (/)  e.  ~P (/) 
/\  (/)  e.  Fin )
)
85, 6, 7mpbir2an 886 . . . . . 6  |-  (/)  e.  ( ~P (/)  i^i  Fin )
9 0iun 3975 . . . . . 6  |-  U_ x  e.  (/)  ( x (
ball `  M )
r )  =  (/)
10 iuneq1 3934 . . . . . . . 8  |-  ( v  =  (/)  ->  U_ x  e.  v  ( x
( ball `  M )
r )  =  U_ x  e.  (/)  ( x ( ball `  M
) r ) )
1110eqeq1d 2304 . . . . . . 7  |-  ( v  =  (/)  ->  ( U_ x  e.  v  (
x ( ball `  M
) r )  =  (/) 
<-> 
U_ x  e.  (/)  ( x ( ball `  M ) r )  =  (/) ) )
1211rspcev 2897 . . . . . 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 2623 . . . 4  |-  A. r  e.  RR+  E. v  e.  ( ~P (/)  i^i  Fin ) U_ x  e.  v  ( x ( ball `  M ) r )  =  (/)
15 istotbnd3 26598 . . . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164   (/)c0 3468   ~Pcpw 3638   U_ciun 3921   ` cfv 5271  (class class class)co 5874   Fincfn 6879   RR+crp 10370   Metcme 16386   ballcbl 16387   TotBndctotbnd 26593
This theorem is referenced by:  prdsbnd2  26622
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
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-ral 2561  df-rex 2562  df-reu 2563  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-int 3879  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-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-fin 6883  df-totbnd 26595
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