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Theorem bndss 28682
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 19936 . . . 4  |-  ( ( M  e.  ( Met `  X )  /\  S  C_  X )  ->  ( M  |`  ( S  X.  S ) )  e.  ( Met `  S
) )
21adantlr 714 . . 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 3349 . . . . . . . . . . . . 13  |-  ( ( S  C_  X  /\  x  e.  S )  ->  x  e.  X )
43ancoms 453 . . . . . . . . . . . 12  |-  ( ( x  e.  S  /\  S  C_  X )  ->  x  e.  X )
5 oveq1 6096 . . . . . . . . . . . . . . 15  |-  ( y  =  x  ->  (
y ( ball `  M
) r )  =  ( x ( ball `  M ) r ) )
65eqeq2d 2452 . . . . . . . . . . . . . 14  |-  ( y  =  x  ->  ( X  =  ( y
( ball `  M )
r )  <->  X  =  ( x ( ball `  M ) r ) ) )
76rexbidv 2734 . . . . . . . . . . . . 13  |-  ( y  =  x  ->  ( E. r  e.  RR+  X  =  ( y ( ball `  M ) r )  <->  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) ) )
87rspcva 3069 . . . . . . . . . . . 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 471 . . . . . . . . . . 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 717 . . . . . . . . . 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 3341 . . . . . . . . . . . . . . . . . . 19  |-  ( S 
C_  X  <->  S  =  ( S  i^i  X ) )
1211biimpi 194 . . . . . . . . . . . . . . . . . 18  |-  ( S 
C_  X  ->  S  =  ( S  i^i  X ) )
13 incom 3541 . . . . . . . . . . . . . . . . . 18  |-  ( S  i^i  X )  =  ( X  i^i  S
)
1412, 13syl6eq 2489 . . . . . . . . . . . . . . . . 17  |-  ( S 
C_  X  ->  S  =  ( X  i^i  S ) )
15 ineq1 3543 . . . . . . . . . . . . . . . . 17  |-  ( X  =  ( x (
ball `  M )
r )  ->  ( X  i^i  S )  =  ( ( x (
ball `  M )
r )  i^i  S
) )
1614, 15sylan9eq 2493 . . . . . . . . . . . . . . . 16  |-  ( ( S  C_  X  /\  X  =  ( x
( ball `  M )
r ) )  ->  S  =  ( (
x ( ball `  M
) r )  i^i 
S ) )
1716adantll 713 . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . 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 2441 . . . . . . . . . . . . . . . . . 18  |-  ( M  |`  ( S  X.  S
) )  =  ( M  |`  ( S  X.  S ) )
2019blssp 28649 . . . . . . . . . . . . . . . . 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 822 . . . . . . . . . . . . . . . 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 648 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . 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 434 . . . . . . . . . . . 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 2826 . . . . . . . . . . 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 429 . . . . . . . . . 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 470 . . . . . . . . 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 802 . . . . . . . 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 434 . . . . . . 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 802 . . . . . 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 429 . . . . 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 802 . . . 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 2797 . . 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 532 . 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 28676 . . 3  |-  ( M  e.  ( Bnd `  X
)  <->  ( M  e.  ( Met `  X
)  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M
) r ) ) )
3736anbi1i 695 . 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 28676 . 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 266 1  |-  ( ( M  e.  ( Bnd `  X )  /\  S  C_  X )  ->  ( M  |`  ( S  X.  S ) )  e.  ( Bnd `  S
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2713   E.wrex 2714    i^i cin 3325    C_ wss 3326    X. cxp 4836    |` cres 4840   ` cfv 5416  (class class class)co 6089   RR+crp 10989   Metcme 17800   ballcbl 17801   Bndcbnd 28663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-mulcl 9342  ax-i2m1 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-rp 10990  df-xadd 11088  df-psmet 17807  df-xmet 17808  df-met 17809  df-bl 17810  df-bnd 28675
This theorem is referenced by:  ssbnd  28684
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