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Theorem bndss 26495
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 18393 . . . 4  |-  ( ( M  e.  ( Met `  X )  /\  S  C_  X )  ->  ( M  |`  ( S  X.  S ) )  e.  ( Met `  S
) )
21adantlr 696 . . 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 3343 . . . . . . . . . . . . 13  |-  ( ( S  C_  X  /\  x  e.  S )  ->  x  e.  X )
43ancoms 440 . . . . . . . . . . . 12  |-  ( ( x  e.  S  /\  S  C_  X )  ->  x  e.  X )
5 oveq1 6088 . . . . . . . . . . . . . . 15  |-  ( y  =  x  ->  (
y ( ball `  M
) r )  =  ( x ( ball `  M ) r ) )
65eqeq2d 2447 . . . . . . . . . . . . . 14  |-  ( y  =  x  ->  ( X  =  ( y
( ball `  M )
r )  <->  X  =  ( x ( ball `  M ) r ) ) )
76rexbidv 2726 . . . . . . . . . . . . 13  |-  ( y  =  x  ->  ( E. r  e.  RR+  X  =  ( y ( ball `  M ) r )  <->  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) ) )
87rspcva 3050 . . . . . . . . . . . 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 458 . . . . . . . . . . 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 699 . . . . . . . . . 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 3335 . . . . . . . . . . . . . . . . . . 19  |-  ( S 
C_  X  <->  S  =  ( S  i^i  X ) )
1211biimpi 187 . . . . . . . . . . . . . . . . . 18  |-  ( S 
C_  X  ->  S  =  ( S  i^i  X ) )
13 incom 3533 . . . . . . . . . . . . . . . . . 18  |-  ( S  i^i  X )  =  ( X  i^i  S
)
1412, 13syl6eq 2484 . . . . . . . . . . . . . . . . 17  |-  ( S 
C_  X  ->  S  =  ( X  i^i  S ) )
15 ineq1 3535 . . . . . . . . . . . . . . . . 17  |-  ( X  =  ( x (
ball `  M )
r )  ->  ( X  i^i  S )  =  ( ( x (
ball `  M )
r )  i^i  S
) )
1614, 15sylan9eq 2488 . . . . . . . . . . . . . . . 16  |-  ( ( S  C_  X  /\  X  =  ( x
( ball `  M )
r ) )  ->  S  =  ( (
x ( ball `  M
) r )  i^i 
S ) )
1716adantll 695 . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . . . . . . 18  |-  ( M  |`  ( S  X.  S
) )  =  ( M  |`  ( S  X.  S ) )
2019blssp 26462 . . . . . . . . . . . . . . . . 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 800 . . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . 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 2471 . . . . . . . . . . . . 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 424 . . . . . . . . . . . 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 2818 . . . . . . . . . . 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 419 . . . . . . . . . 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 457 . . . . . . . . 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 780 . . . . . . . 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 424 . . . . . . 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 780 . . . . . 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 419 . . . . 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 780 . . . 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 2789 . . 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 519 . 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 26489 . . 3  |-  ( M  e.  ( Bnd `  X
)  <->  ( M  e.  ( Met `  X
)  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M
) r ) ) )
3736anbi1i 677 . 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 26489 . 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 258 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 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    i^i cin 3319    C_ wss 3320    X. cxp 4876    |` cres 4880   ` cfv 5454  (class class class)co 6081   RR+crp 10612   Metcme 16687   ballcbl 16688   Bndcbnd 26476
This theorem is referenced by:  ssbnd  26497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-mulcl 9052  ax-i2m1 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-rp 10613  df-xadd 10711  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-bnd 26488
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