Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bndss Unicode version

Theorem bndss 26613
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 17943 . . . 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 3188 . . . . . . . . . . . . 13  |-  ( ( S  C_  X  /\  x  e.  S )  ->  x  e.  X )
43ancoms 439 . . . . . . . . . . . 12  |-  ( ( x  e.  S  /\  S  C_  X )  ->  x  e.  X )
5 oveq1 5881 . . . . . . . . . . . . . . 15  |-  ( y  =  x  ->  (
y ( ball `  M
) r )  =  ( x ( ball `  M ) r ) )
65eqeq2d 2307 . . . . . . . . . . . . . 14  |-  ( y  =  x  ->  ( X  =  ( y
( ball `  M )
r )  <->  X  =  ( x ( ball `  M ) r ) ) )
76rexbidv 2577 . . . . . . . . . . . . 13  |-  ( y  =  x  ->  ( E. r  e.  RR+  X  =  ( y ( ball `  M ) r )  <->  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) ) )
87rspcva 2895 . . . . . . . . . . . 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 3180 . . . . . . . . . . . . . . . . . . 19  |-  ( S 
C_  X  <->  S  =  ( S  i^i  X ) )
1211biimpi 186 . . . . . . . . . . . . . . . . . 18  |-  ( S 
C_  X  ->  S  =  ( S  i^i  X ) )
13 incom 3374 . . . . . . . . . . . . . . . . . 18  |-  ( S  i^i  X )  =  ( X  i^i  S
)
1412, 13syl6eq 2344 . . . . . . . . . . . . . . . . 17  |-  ( S 
C_  X  ->  S  =  ( X  i^i  S ) )
15 ineq1 3376 . . . . . . . . . . . . . . . . 17  |-  ( X  =  ( x (
ball `  M )
r )  ->  ( X  i^i  S )  =  ( ( x (
ball `  M )
r )  i^i  S
) )
1614, 15sylan9eq 2348 . . . . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . . . . 18  |-  ( M  |`  ( S  X.  S
) )  =  ( M  |`  ( S  X.  S ) )
2019blssp 26573 . . . . . . . . . . . . . . . . 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 2331 . . . . . . . . . . . . 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 2668 . . . . . . . . . . 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 2639 . . 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 26607 . . 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 26607 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165    X. cxp 4703    |` cres 4707   ` cfv 5271  (class class class)co 5874   RR+crp 10370   Metcme 16386   ballcbl 16387   Bndcbnd 26594
This theorem is referenced by:  ssbnd  26615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-mulcl 8815  ax-i2m1 8821
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-rp 10371  df-xadd 10469  df-xmet 16389  df-met 16390  df-bl 16391  df-bnd 26606
  Copyright terms: Public domain W3C validator