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Theorem totbndss 28520
Description: A subset of a totally bounded metric space is totally bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
totbndss  |-  ( ( M  e.  ( TotBnd `  X )  /\  S  C_  X )  ->  ( M  |`  ( S  X.  S ) )  e.  ( TotBnd `  S )
)

Proof of Theorem totbndss
Dummy variables  b 
d  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istotbnd 28512 . . . 4  |-  ( M  e.  ( TotBnd `  X
)  <->  ( M  e.  ( Met `  X
)  /\  A. d  e.  RR+  E. v  e. 
Fin  ( U. v  =  X  /\  A. b  e.  v  E. x  e.  X  b  =  ( x ( ball `  M ) d ) ) ) )
21simprbi 461 . . 3  |-  ( M  e.  ( TotBnd `  X
)  ->  A. d  e.  RR+  E. v  e. 
Fin  ( U. v  =  X  /\  A. b  e.  v  E. x  e.  X  b  =  ( x ( ball `  M ) d ) ) )
3 sseq2 3366 . . . . . . 7  |-  ( U. v  =  X  ->  ( S  C_  U. v  <->  S 
C_  X ) )
43biimprcd 225 . . . . . 6  |-  ( S 
C_  X  ->  ( U. v  =  X  ->  S  C_  U. v
) )
54anim1d 559 . . . . 5  |-  ( S 
C_  X  ->  (
( U. v  =  X  /\  A. b  e.  v  E. x  e.  X  b  =  ( x ( ball `  M ) d ) )  ->  ( S  C_ 
U. v  /\  A. b  e.  v  E. x  e.  X  b  =  ( x (
ball `  M )
d ) ) ) )
65reximdv 2817 . . . 4  |-  ( S 
C_  X  ->  ( E. v  e.  Fin  ( U. v  =  X  /\  A. b  e.  v  E. x  e.  X  b  =  ( x ( ball `  M
) d ) )  ->  E. v  e.  Fin  ( S  C_  U. v  /\  A. b  e.  v  E. x  e.  X  b  =  ( x
( ball `  M )
d ) ) ) )
76ralimdv 2785 . . 3  |-  ( S 
C_  X  ->  ( A. d  e.  RR+  E. v  e.  Fin  ( U. v  =  X  /\  A. b  e.  v  E. x  e.  X  b  =  ( x ( ball `  M ) d ) )  ->  A. d  e.  RR+  E. v  e. 
Fin  ( S  C_  U. v  /\  A. b  e.  v  E. x  e.  X  b  =  ( x ( ball `  M ) d ) ) ) )
82, 7mpan9 466 . 2  |-  ( ( M  e.  ( TotBnd `  X )  /\  S  C_  X )  ->  A. d  e.  RR+  E. v  e. 
Fin  ( S  C_  U. v  /\  A. b  e.  v  E. x  e.  X  b  =  ( x ( ball `  M ) d ) ) )
9 totbndmet 28515 . . 3  |-  ( M  e.  ( TotBnd `  X
)  ->  M  e.  ( Met `  X ) )
10 eqid 2433 . . . 4  |-  ( M  |`  ( S  X.  S
) )  =  ( M  |`  ( S  X.  S ) )
1110sstotbnd 28518 . . 3  |-  ( ( M  e.  ( Met `  X )  /\  S  C_  X )  ->  (
( M  |`  ( S  X.  S ) )  e.  ( TotBnd `  S
)  <->  A. d  e.  RR+  E. v  e.  Fin  ( S  C_  U. v  /\  A. b  e.  v  E. x  e.  X  b  =  ( x (
ball `  M )
d ) ) ) )
129, 11sylan 468 . 2  |-  ( ( M  e.  ( TotBnd `  X )  /\  S  C_  X )  ->  (
( M  |`  ( S  X.  S ) )  e.  ( TotBnd `  S
)  <->  A. d  e.  RR+  E. v  e.  Fin  ( S  C_  U. v  /\  A. b  e.  v  E. x  e.  X  b  =  ( x (
ball `  M )
d ) ) ) )
138, 12mpbird 232 1  |-  ( ( M  e.  ( TotBnd `  X )  /\  S  C_  X )  ->  ( M  |`  ( S  X.  S ) )  e.  ( TotBnd `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362    e. wcel 1755   A.wral 2705   E.wrex 2706    C_ wss 3316   U.cuni 4079    X. cxp 4825    |` cres 4829   ` cfv 5406  (class class class)co 6080   Fincfn 7298   RR+crp 10979   Metcme 17646   ballcbl 17647   TotBndctotbnd 28509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-2 10368  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-totbnd 28511
This theorem is referenced by:  prdsbnd2  28538
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