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Theorem totbndss 25913
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 25905 . . . 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 450 . . 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 3200 . . . . . . 7  |-  ( U. v  =  X  ->  ( S  C_  U. v  <->  S 
C_  X ) )
43biimprcd 216 . . . . . 6  |-  ( S 
C_  X  ->  ( U. v  =  X  ->  S  C_  U. v
) )
54anim1d 547 . . . . 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 2654 . . . 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 2622 . . 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 455 . 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 25908 . . 3  |-  ( M  e.  ( TotBnd `  X
)  ->  M  e.  ( Met `  X ) )
10 eqid 2283 . . . 4  |-  ( M  |`  ( S  X.  S
) )  =  ( M  |`  ( S  X.  S ) )
1110sstotbnd 25911 . . 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 457 . 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 223 1  |-  ( ( M  e.  ( TotBnd `  X )  /\  S  C_  X )  ->  ( M  |`  ( S  X.  S ) )  e.  ( TotBnd `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   U.cuni 3827    X. cxp 4687    |` cres 4691   ` cfv 5255  (class class class)co 5858   Fincfn 6863   RR+crp 10354   Metcme 16370   ballcbl 16371   TotBndctotbnd 25902
This theorem is referenced by:  prdsbnd2  25931
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  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-xmet 16373  df-met 16374  df-bl 16375  df-totbnd 25904
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