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Theorem bndss 26510
Description: A subset of a bounded metric space is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
bndss  |-  ( ( M  e.  ( Bnd `  X )  /\  S  C_  X )  ->  ( M  |`  ( S  X.  S ) )  e.  ( Bnd `  S
) )

Proof of Theorem bndss
Dummy variables  r  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 metres2 17927 . . . 4  |-  ( ( M  e.  ( Met `  X )  /\  S  C_  X )  ->  ( M  |`  ( S  X.  S ) )  e.  ( Met `  S
) )
21adantlr 695 . . 3  |-  ( ( ( M  e.  ( Met `  X )  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M
) r ) )  /\  S  C_  X
)  ->  ( M  |`  ( S  X.  S
) )  e.  ( Met `  S ) )
3 ssel2 3175 . . . . . . . . . . . . 13  |-  ( ( S  C_  X  /\  x  e.  S )  ->  x  e.  X )
43ancoms 439 . . . . . . . . . . . 12  |-  ( ( x  e.  S  /\  S  C_  X )  ->  x  e.  X )
5 oveq1 5865 . . . . . . . . . . . . . . 15  |-  ( y  =  x  ->  (
y ( ball `  M
) r )  =  ( x ( ball `  M ) r ) )
65eqeq2d 2294 . . . . . . . . . . . . . 14  |-  ( y  =  x  ->  ( X  =  ( y
( ball `  M )
r )  <->  X  =  ( x ( ball `  M ) r ) ) )
76rexbidv 2564 . . . . . . . . . . . . 13  |-  ( y  =  x  ->  ( E. r  e.  RR+  X  =  ( y ( ball `  M ) r )  <->  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) ) )
87rspcva 2882 . . . . . . . . . . . 12  |-  ( ( x  e.  X  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M ) r ) )  ->  E. r  e.  RR+  X  =  ( x ( ball `  M
) r ) )
94, 8sylan 457 . . . . . . . . . . 11  |-  ( ( ( x  e.  S  /\  S  C_  X )  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M
) r ) )  ->  E. r  e.  RR+  X  =  ( x (
ball `  M )
r ) )
109adantlll 698 . . . . . . . . . 10  |-  ( ( ( ( M  e.  ( Met `  X
)  /\  x  e.  S )  /\  S  C_  X )  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M ) r ) )  ->  E. r  e.  RR+  X  =  ( x ( ball `  M
) r ) )
11 dfss 3167 . . . . . . . . . . . . . . . . . . 19  |-  ( S 
C_  X  <->  S  =  ( S  i^i  X ) )
1211biimpi 186 . . . . . . . . . . . . . . . . . 18  |-  ( S 
C_  X  ->  S  =  ( S  i^i  X ) )
13 incom 3361 . . . . . . . . . . . . . . . . . 18  |-  ( S  i^i  X )  =  ( X  i^i  S
)
1412, 13syl6eq 2331 . . . . . . . . . . . . . . . . 17  |-  ( S 
C_  X  ->  S  =  ( X  i^i  S ) )
15 ineq1 3363 . . . . . . . . . . . . . . . . 17  |-  ( X  =  ( x (
ball `  M )
r )  ->  ( X  i^i  S )  =  ( ( x (
ball `  M )
r )  i^i  S
) )
1614, 15sylan9eq 2335 . . . . . . . . . . . . . . . 16  |-  ( ( S  C_  X  /\  X  =  ( x
( ball `  M )
r ) )  ->  S  =  ( (
x ( ball `  M
) r )  i^i 
S ) )
1716adantll 694 . . . . . . . . . . . . . . 15  |-  ( ( ( ( M  e.  ( Met `  X
)  /\  x  e.  S )  /\  S  C_  X )  /\  X  =  ( x (
ball `  M )
r ) )  ->  S  =  ( (
x ( ball `  M
) r )  i^i 
S ) )
1817adantlr 695 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( M  e.  ( Met `  X
)  /\  x  e.  S )  /\  S  C_  X )  /\  r  e.  RR+ )  /\  X  =  ( x (
ball `  M )
r ) )  ->  S  =  ( (
x ( ball `  M
) r )  i^i 
S ) )
19 eqid 2283 . . . . . . . . . . . . . . . . . 18  |-  ( M  |`  ( S  X.  S
) )  =  ( M  |`  ( S  X.  S ) )
2019blssp 26470 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M  e.  ( Met `  X )  /\  S  C_  X
)  /\  ( x  e.  S  /\  r  e.  RR+ ) )  -> 
( x ( ball `  ( M  |`  ( S  X.  S ) ) ) r )  =  ( ( x (
ball `  M )
r )  i^i  S
) )
2120an4s 799 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ( Met `  X )  /\  x  e.  S
)  /\  ( S  C_  X  /\  r  e.  RR+ ) )  ->  (
x ( ball `  ( M  |`  ( S  X.  S ) ) ) r )  =  ( ( x ( ball `  M ) r )  i^i  S ) )
2221anassrs 629 . . . . . . . . . . . . . . 15  |-  ( ( ( ( M  e.  ( Met `  X
)  /\  x  e.  S )  /\  S  C_  X )  /\  r  e.  RR+ )  ->  (
x ( ball `  ( M  |`  ( S  X.  S ) ) ) r )  =  ( ( x ( ball `  M ) r )  i^i  S ) )
2322adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( M  e.  ( Met `  X
)  /\  x  e.  S )  /\  S  C_  X )  /\  r  e.  RR+ )  /\  X  =  ( x (
ball `  M )
r ) )  -> 
( x ( ball `  ( M  |`  ( S  X.  S ) ) ) r )  =  ( ( x (
ball `  M )
r )  i^i  S
) )
2418, 23eqtr4d 2318 . . . . . . . . . . . . 13  |-  ( ( ( ( ( M  e.  ( Met `  X
)  /\  x  e.  S )  /\  S  C_  X )  /\  r  e.  RR+ )  /\  X  =  ( x (
ball `  M )
r ) )  ->  S  =  ( x
( ball `  ( M  |`  ( S  X.  S
) ) ) r ) )
2524ex 423 . . . . . . . . . . . 12  |-  ( ( ( ( M  e.  ( Met `  X
)  /\  x  e.  S )  /\  S  C_  X )  /\  r  e.  RR+ )  ->  ( X  =  ( x
( ball `  M )
r )  ->  S  =  ( x (
ball `  ( M  |`  ( S  X.  S
) ) ) r ) ) )
2625reximdva 2655 . . . . . . . . . . 11  |-  ( ( ( M  e.  ( Met `  X )  /\  x  e.  S
)  /\  S  C_  X
)  ->  ( E. r  e.  RR+  X  =  ( x ( ball `  M ) r )  ->  E. r  e.  RR+  S  =  ( x (
ball `  ( M  |`  ( S  X.  S
) ) ) r ) ) )
2726imp 418 . . . . . . . . . 10  |-  ( ( ( ( M  e.  ( Met `  X
)  /\  x  e.  S )  /\  S  C_  X )  /\  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) )  ->  E. r  e.  RR+  S  =  ( x ( ball `  ( M  |`  ( S  X.  S ) ) ) r ) )
2810, 27syldan 456 . . . . . . . . 9  |-  ( ( ( ( M  e.  ( Met `  X
)  /\  x  e.  S )  /\  S  C_  X )  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M ) r ) )  ->  E. r  e.  RR+  S  =  ( x ( ball `  ( M  |`  ( S  X.  S ) ) ) r ) )
2928an32s 779 . . . . . . . 8  |-  ( ( ( ( M  e.  ( Met `  X
)  /\  x  e.  S )  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M ) r ) )  /\  S  C_  X )  ->  E. r  e.  RR+  S  =  ( x ( ball `  ( M  |`  ( S  X.  S ) ) ) r ) )
3029ex 423 . . . . . . 7  |-  ( ( ( M  e.  ( Met `  X )  /\  x  e.  S
)  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M
) r ) )  ->  ( S  C_  X  ->  E. r  e.  RR+  S  =  ( x (
ball `  ( M  |`  ( S  X.  S
) ) ) r ) ) )
3130an32s 779 . . . . . 6  |-  ( ( ( M  e.  ( Met `  X )  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M
) r ) )  /\  x  e.  S
)  ->  ( S  C_  X  ->  E. r  e.  RR+  S  =  ( x ( ball `  ( M  |`  ( S  X.  S ) ) ) r ) ) )
3231imp 418 . . . . 5  |-  ( ( ( ( M  e.  ( Met `  X
)  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M
) r ) )  /\  x  e.  S
)  /\  S  C_  X
)  ->  E. r  e.  RR+  S  =  ( x ( ball `  ( M  |`  ( S  X.  S ) ) ) r ) )
3332an32s 779 . . . 4  |-  ( ( ( ( M  e.  ( Met `  X
)  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M
) r ) )  /\  S  C_  X
)  /\  x  e.  S )  ->  E. r  e.  RR+  S  =  ( x ( ball `  ( M  |`  ( S  X.  S ) ) ) r ) )
3433ralrimiva 2626 . . 3  |-  ( ( ( M  e.  ( Met `  X )  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M
) r ) )  /\  S  C_  X
)  ->  A. x  e.  S  E. r  e.  RR+  S  =  ( x ( ball `  ( M  |`  ( S  X.  S ) ) ) r ) )
352, 34jca 518 . 2  |-  ( ( ( M  e.  ( Met `  X )  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M
) r ) )  /\  S  C_  X
)  ->  ( ( M  |`  ( S  X.  S ) )  e.  ( Met `  S
)  /\  A. x  e.  S  E. r  e.  RR+  S  =  ( x ( ball `  ( M  |`  ( S  X.  S ) ) ) r ) ) )
36 isbnd 26504 . . 3  |-  ( M  e.  ( Bnd `  X
)  <->  ( M  e.  ( Met `  X
)  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M
) r ) ) )
3736anbi1i 676 . 2  |-  ( ( M  e.  ( Bnd `  X )  /\  S  C_  X )  <->  ( ( M  e.  ( Met `  X )  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M ) r ) )  /\  S  C_  X ) )
38 isbnd 26504 . 2  |-  ( ( M  |`  ( S  X.  S ) )  e.  ( Bnd `  S
)  <->  ( ( M  |`  ( S  X.  S
) )  e.  ( Met `  S )  /\  A. x  e.  S  E. r  e.  RR+  S  =  ( x ( ball `  ( M  |`  ( S  X.  S ) ) ) r ) ) )
3935, 37, 383imtr4i 257 1  |-  ( ( M  e.  ( Bnd `  X )  /\  S  C_  X )  ->  ( M  |`  ( S  X.  S ) )  e.  ( Bnd `  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152    X. cxp 4687    |` cres 4691   ` cfv 5255  (class class class)co 5858   RR+crp 10354   Metcme 16370   ballcbl 16371   Bndcbnd 26491
This theorem is referenced by:  ssbnd  26512
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-mulcl 8799  ax-i2m1 8805
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-rp 10355  df-xadd 10453  df-xmet 16373  df-met 16374  df-bl 16375  df-bnd 26503
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