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Theorem bndss 33585
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 22168 . . . 4  |-  ( ( M  e.  ( Met `  X )  /\  S  C_  X )  ->  ( M  |`  ( S  X.  S ) )  e.  ( Met `  S
) )
21adantlr 751 . . 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 3598 . . . . . . . . . . . . 13  |-  ( ( S  C_  X  /\  x  e.  S )  ->  x  e.  X )
43ancoms 469 . . . . . . . . . . . 12  |-  ( ( x  e.  S  /\  S  C_  X )  ->  x  e.  X )
5 oveq1 6657 . . . . . . . . . . . . . . 15  |-  ( y  =  x  ->  (
y ( ball `  M
) r )  =  ( x ( ball `  M ) r ) )
65eqeq2d 2632 . . . . . . . . . . . . . 14  |-  ( y  =  x  ->  ( X  =  ( y
( ball `  M )
r )  <->  X  =  ( x ( ball `  M ) r ) ) )
76rexbidv 3052 . . . . . . . . . . . . 13  |-  ( y  =  x  ->  ( E. r  e.  RR+  X  =  ( y ( ball `  M ) r )  <->  E. r  e.  RR+  X  =  ( x ( ball `  M ) r ) ) )
87rspcva 3307 . . . . . . . . . . . 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 488 . . . . . . . . . . 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 754 . . . . . . . . . 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 3589 . . . . . . . . . . . . . . . . . . 19  |-  ( S 
C_  X  <->  S  =  ( S  i^i  X ) )
1211biimpi 206 . . . . . . . . . . . . . . . . . 18  |-  ( S 
C_  X  ->  S  =  ( S  i^i  X ) )
13 incom 3805 . . . . . . . . . . . . . . . . . 18  |-  ( S  i^i  X )  =  ( X  i^i  S
)
1412, 13syl6eq 2672 . . . . . . . . . . . . . . . . 17  |-  ( S 
C_  X  ->  S  =  ( X  i^i  S ) )
15 ineq1 3807 . . . . . . . . . . . . . . . . 17  |-  ( X  =  ( x (
ball `  M )
r )  ->  ( X  i^i  S )  =  ( ( x (
ball `  M )
r )  i^i  S
) )
1614, 15sylan9eq 2676 . . . . . . . . . . . . . . . 16  |-  ( ( S  C_  X  /\  X  =  ( x
( ball `  M )
r ) )  ->  S  =  ( (
x ( ball `  M
) r )  i^i 
S ) )
1716adantll 750 . . . . . . . . . . . . . . 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 751 . . . . . . . . . . . . . 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 2622 . . . . . . . . . . . . . . . . . 18  |-  ( M  |`  ( S  X.  S
) )  =  ( M  |`  ( S  X.  S ) )
2019blssp 33552 . . . . . . . . . . . . . . . . 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 869 . . . . . . . . . . . . . . . 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 680 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 2659 . . . . . . . . . . . . 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 450 . . . . . . . . . . . 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 3017 . . . . . . . . . . 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 445 . . . . . . . . . 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 487 . . . . . . . . 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 846 . . . . . . . 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 450 . . . . . . 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 846 . . . . . 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 445 . . . . 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 846 . . . 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 2966 . . 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 554 . 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 33579 . . 3  |-  ( M  e.  ( Bnd `  X
)  <->  ( M  e.  ( Met `  X
)  /\  A. y  e.  X  E. r  e.  RR+  X  =  ( y ( ball `  M
) r ) ) )
3736anbi1i 731 . 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 33579 . 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 281 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 384    = wceq 1483    e. wcel 1990   A.wral 2912   E.wrex 2913    i^i cin 3573    C_ wss 3574    X. cxp 5112    |` cres 5116   ` cfv 5888  (class class class)co 6650   RR+crp 11832   Metcme 19732   ballcbl 19733   Bndcbnd 33566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-rp 11833  df-xadd 11947  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-bnd 33578
This theorem is referenced by:  ssbnd  33587
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