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Theorem reheibor 25886
Description: Heine-Borel theorem for real numbers. A subset of  RR is compact iff it is closed and bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
reheibor.2  |-  M  =  ( ( abs  o.  -  )  |`  ( Y  X.  Y ) )
reheibor.3  |-  T  =  ( MetOpen `  M )
reheibor.4  |-  U  =  ( topGen `  ran  (,) )
Assertion
Ref Expression
reheibor  |-  ( Y 
C_  RR  ->  ( T  e.  Comp  <->  ( Y  e.  ( Clsd `  U
)  /\  M  e.  ( Bnd `  Y ) ) ) )

Proof of Theorem reheibor
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df1o2 6578 . . . 4  |-  1o  =  { (/) }
2 snfi 7029 . . . 4  |-  { (/) }  e.  Fin
31, 2eqeltri 2428 . . 3  |-  1o  e.  Fin
4 imassrn 5107 . . . . 5  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  C_  ran  ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
5 0ex 4231 . . . . . . . . . 10  |-  (/)  e.  _V
6 eqid 2358 . . . . . . . . . . 11  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
7 eqid 2358 . . . . . . . . . . 11  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  =  ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
86, 7ismrer1 25885 . . . . . . . . . 10  |-  ( (/)  e.  _V  ->  ( x  e.  RR  |->  ( { (/) }  X.  { x }
) )  e.  ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) 
Ismty  ( Rn `  { (/)
} ) ) )
95, 8ax-mp 8 . . . . . . . . 9  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  e.  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) 
Ismty  ( Rn `  { (/)
} ) )
101fveq2i 5611 . . . . . . . . . 10  |-  ( Rn
`  1o )  =  ( Rn `  { (/)
} )
1110oveq2i 5956 . . . . . . . . 9  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  1o ) )  =  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  { (/) } ) )
129, 11eleqtrri 2431 . . . . . . . 8  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  e.  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) 
Ismty  ( Rn `  1o ) )
136rexmet 18399 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
14 eqid 2358 . . . . . . . . . . 11  |-  ( RR 
^m  1o )  =  ( RR  ^m  1o )
1514rrnmet 25876 . . . . . . . . . 10  |-  ( 1o  e.  Fin  ->  ( Rn `  1o )  e.  ( Met `  ( RR  ^m  1o ) ) )
16 metxmet 18001 . . . . . . . . . 10  |-  ( ( Rn `  1o )  e.  ( Met `  ( RR  ^m  1o ) )  ->  ( Rn `  1o )  e.  ( * Met `  ( RR 
^m  1o ) ) )
173, 15, 16mp2b 9 . . . . . . . . 9  |-  ( Rn
`  1o )  e.  ( * Met `  ( RR  ^m  1o ) )
18 isismty 25848 . . . . . . . . 9  |-  ( ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  e.  ( * Met `  RR )  /\  ( Rn `  1o )  e.  ( * Met `  ( RR  ^m  1o ) ) )  ->  ( (
x  e.  RR  |->  ( { (/) }  X.  {
x } ) )  e.  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  1o ) )  <-> 
( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) : RR -1-1-onto-> ( RR  ^m  1o )  /\  A. y  e.  RR  A. z  e.  RR  (
y ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) z )  =  ( ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) `  y
) ( Rn `  1o ) ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) `  z ) ) ) ) )
1913, 17, 18mp2an 653 . . . . . . . 8  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )  e.  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  1o ) )  <-> 
( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) : RR -1-1-onto-> ( RR  ^m  1o )  /\  A. y  e.  RR  A. z  e.  RR  (
y ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) z )  =  ( ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) `  y
) ( Rn `  1o ) ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) `  z ) ) ) )
2012, 19mpbi 199 . . . . . . 7  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) ) : RR -1-1-onto-> ( RR  ^m  1o )  /\  A. y  e.  RR  A. z  e.  RR  ( y ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) z )  =  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) ) `
 y ) ( Rn `  1o ) ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) `  z
) ) )
2120simpli 444 . . . . . 6  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) : RR -1-1-onto-> ( RR  ^m  1o )
22 f1of 5555 . . . . . 6  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) ) : RR -1-1-onto-> ( RR  ^m  1o )  ->  ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) : RR --> ( RR  ^m  1o ) )
23 frn 5478 . . . . . 6  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) ) : RR --> ( RR 
^m  1o )  ->  ran  ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) ) 
C_  ( RR  ^m  1o ) )
2421, 22, 23mp2b 9 . . . . 5  |-  ran  (
x  e.  RR  |->  ( { (/) }  X.  {
x } ) ) 
C_  ( RR  ^m  1o )
254, 24sstri 3264 . . . 4  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  C_  ( RR  ^m  1o )
2625a1i 10 . . 3  |-  ( Y 
C_  RR  ->  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  C_  ( RR  ^m  1o ) )
27 eqid 2358 . . . 4  |-  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) )  =  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) )
28 eqid 2358 . . . 4  |-  ( MetOpen `  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) )  =  ( MetOpen `  (
( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) )
29 eqid 2358 . . . 4  |-  ( MetOpen `  ( Rn `  1o ) )  =  ( MetOpen `  ( Rn `  1o ) )
3014, 27, 28, 29rrnheibor 25884 . . 3  |-  ( ( 1o  e.  Fin  /\  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
)  C_  ( RR  ^m  1o ) )  -> 
( ( MetOpen `  (
( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) )  e.  Comp  <->  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  e.  ( Clsd `  ( MetOpen
`  ( Rn `  1o ) ) )  /\  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) )  e.  ( Bnd `  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) ) )
313, 26, 30sylancr 644 . 2  |-  ( Y 
C_  RR  ->  ( (
MetOpen `  ( ( Rn
`  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) )  e.  Comp  <->  (
( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
)  e.  ( Clsd `  ( MetOpen `  ( Rn `  1o ) ) )  /\  ( ( Rn
`  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) )  e.  ( Bnd `  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) ) )
32 reheibor.2 . . . . . . 7  |-  M  =  ( ( abs  o.  -  )  |`  ( Y  X.  Y ) )
33 cnxmet 18384 . . . . . . . 8  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
34 id 19 . . . . . . . . 9  |-  ( Y 
C_  RR  ->  Y  C_  RR )
35 ax-resscn 8884 . . . . . . . . 9  |-  RR  C_  CC
3634, 35syl6ss 3267 . . . . . . . 8  |-  ( Y 
C_  RR  ->  Y  C_  CC )
37 xmetres2 18027 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  e.  ( * Met `  CC )  /\  Y  C_  CC )  -> 
( ( abs  o.  -  )  |`  ( Y  X.  Y ) )  e.  ( * Met `  Y ) )
3833, 36, 37sylancr 644 . . . . . . 7  |-  ( Y 
C_  RR  ->  ( ( abs  o.  -  )  |`  ( Y  X.  Y
) )  e.  ( * Met `  Y
) )
3932, 38syl5eqel 2442 . . . . . 6  |-  ( Y 
C_  RR  ->  M  e.  ( * Met `  Y
) )
40 xmetres2 18027 . . . . . . 7  |-  ( ( ( Rn `  1o )  e.  ( * Met `  ( RR  ^m  1o ) )  /\  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  C_  ( RR  ^m  1o ) )  ->  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) )  e.  ( * Met `  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) )
4117, 26, 40sylancr 644 . . . . . 6  |-  ( Y 
C_  RR  ->  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) )  e.  ( * Met `  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) )
42 reheibor.3 . . . . . . 7  |-  T  =  ( MetOpen `  M )
4342, 28ismtyhmeo 25852 . . . . . 6  |-  ( ( M  e.  ( * Met `  Y )  /\  ( ( Rn
`  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) )  e.  ( * Met `  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) )  ->  ( M  Ismty  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) ) 
C_  ( T  Homeo  (
MetOpen `  ( ( Rn
`  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) ) ) )
4439, 41, 43syl2anc 642 . . . . 5  |-  ( Y 
C_  RR  ->  ( M 
Ismty  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) ) 
C_  ( T  Homeo  (
MetOpen `  ( ( Rn
`  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) ) ) )
4513a1i 10 . . . . . . 7  |-  ( Y 
C_  RR  ->  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR ) )
4617a1i 10 . . . . . . 7  |-  ( Y 
C_  RR  ->  ( Rn
`  1o )  e.  ( * Met `  ( RR  ^m  1o ) ) )
4712a1i 10 . . . . . . 7  |-  ( Y 
C_  RR  ->  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  e.  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) 
Ismty  ( Rn `  1o ) ) )
48 eqid 2358 . . . . . . . 8  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  =  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
)
49 eqid 2358 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )  =  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )
5048, 49, 27ismtyres 25855 . . . . . . 7  |-  ( ( ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )  /\  ( Rn `  1o )  e.  ( * Met `  ( RR  ^m  1o ) ) )  /\  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )  e.  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  1o ) )  /\  Y  C_  RR ) )  ->  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )  |`  Y )  e.  ( ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y
) )  Ismty  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) ) )
5145, 46, 47, 34, 50syl22anc 1183 . . . . . 6  |-  ( Y 
C_  RR  ->  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )  |`  Y )  e.  ( ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y
) )  Ismty  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) ) )
52 xpss12 4874 . . . . . . . . . 10  |-  ( ( Y  C_  RR  /\  Y  C_  RR )  ->  ( Y  X.  Y )  C_  ( RR  X.  RR ) )
5352anidms 626 . . . . . . . . 9  |-  ( Y 
C_  RR  ->  ( Y  X.  Y )  C_  ( RR  X.  RR ) )
54 resabs1 5066 . . . . . . . . 9  |-  ( ( Y  X.  Y ) 
C_  ( RR  X.  RR )  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )  =  ( ( abs 
o.  -  )  |`  ( Y  X.  Y ) ) )
5553, 54syl 15 . . . . . . . 8  |-  ( Y 
C_  RR  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )  =  ( ( abs 
o.  -  )  |`  ( Y  X.  Y ) ) )
5655, 32syl6eqr 2408 . . . . . . 7  |-  ( Y 
C_  RR  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )  =  M )
5756oveq1d 5960 . . . . . 6  |-  ( Y 
C_  RR  ->  ( ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  |`  ( Y  X.  Y
) )  Ismty  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) )  =  ( M  Ismty  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) ) )
5851, 57eleqtrd 2434 . . . . 5  |-  ( Y 
C_  RR  ->  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )  |`  Y )  e.  ( M  Ismty  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) ) )
5944, 58sseldd 3257 . . . 4  |-  ( Y 
C_  RR  ->  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )  |`  Y )  e.  ( T  Homeo  ( MetOpen `  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) ) ) )
60 hmphi 17574 . . . 4  |-  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )  |`  Y )  e.  ( T  Homeo  ( MetOpen `  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) ) )  ->  T  ~=  ( MetOpen `  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) ) )
6159, 60syl 15 . . 3  |-  ( Y 
C_  RR  ->  T  ~=  ( MetOpen `  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) ) )
62 cmphmph 17585 . . . 4  |-  ( T  ~=  ( MetOpen `  (
( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) )  ->  ( T  e. 
Comp  ->  ( MetOpen `  (
( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) )  e.  Comp ) )
63 hmphsym 17579 . . . . 5  |-  ( T  ~=  ( MetOpen `  (
( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) )  ->  ( MetOpen `  (
( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) )  ~=  T )
64 cmphmph 17585 . . . . 5  |-  ( (
MetOpen `  ( ( Rn
`  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) )  ~=  T  ->  ( ( MetOpen `  (
( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) )  e.  Comp  ->  T  e. 
Comp ) )
6563, 64syl 15 . . . 4  |-  ( T  ~=  ( MetOpen `  (
( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) )  ->  ( ( MetOpen `  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) )  e.  Comp  ->  T  e. 
Comp ) )
6662, 65impbid 183 . . 3  |-  ( T  ~=  ( MetOpen `  (
( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) )  ->  ( T  e. 
Comp 
<->  ( MetOpen `  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) )  e.  Comp ) )
6761, 66syl 15 . 2  |-  ( Y 
C_  RR  ->  ( T  e.  Comp  <->  ( MetOpen `  (
( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) )  e.  Comp ) )
68 reheibor.4 . . . . . . . 8  |-  U  =  ( topGen `  ran  (,) )
69 eqid 2358 . . . . . . . . 9  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
706, 69tgioo 18404 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
7168, 70eqtri 2378 . . . . . . 7  |-  U  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
7271, 29ismtyhmeo 25852 . . . . . 6  |-  ( ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  e.  ( * Met `  RR )  /\  ( Rn `  1o )  e.  ( * Met `  ( RR  ^m  1o ) ) )  ->  ( (
( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  1o ) ) 
C_  ( U  Homeo  (
MetOpen `  ( Rn `  1o ) ) ) )
7313, 17, 72mp2an 653 . . . . 5  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  1o ) ) 
C_  ( U  Homeo  (
MetOpen `  ( Rn `  1o ) ) )
7473, 12sselii 3253 . . . 4  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  e.  ( U  Homeo  ( MetOpen `  ( Rn `  1o ) ) )
75 retopon 18374 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
7668, 75eqeltri 2428 . . . . . 6  |-  U  e.  (TopOn `  RR )
7776toponunii 16776 . . . . 5  |-  RR  =  U. U
7877hmeocld 17564 . . . 4  |-  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )  e.  ( U  Homeo  (
MetOpen `  ( Rn `  1o ) ) )  /\  Y  C_  RR )  -> 
( Y  e.  (
Clsd `  U )  <->  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  e.  ( Clsd `  ( MetOpen
`  ( Rn `  1o ) ) ) ) )
7974, 34, 78sylancr 644 . . 3  |-  ( Y 
C_  RR  ->  ( Y  e.  ( Clsd `  U
)  <->  ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  e.  (
Clsd `  ( MetOpen `  ( Rn `  1o ) ) ) ) )
80 ismtybnd 25854 . . . 4  |-  ( ( M  e.  ( * Met `  Y )  /\  ( ( Rn
`  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) )  e.  ( * Met `  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) )  /\  ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  |`  Y )  e.  ( M  Ismty  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) ) )  -> 
( M  e.  ( Bnd `  Y )  <-> 
( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) )  e.  ( Bnd `  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) )
8139, 41, 58, 80syl3anc 1182 . . 3  |-  ( Y 
C_  RR  ->  ( M  e.  ( Bnd `  Y
)  <->  ( ( Rn
`  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) )  e.  ( Bnd `  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) ) )
8279, 81anbi12d 691 . 2  |-  ( Y 
C_  RR  ->  ( ( Y  e.  ( Clsd `  U )  /\  M  e.  ( Bnd `  Y
) )  <->  ( (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  e.  ( Clsd `  ( MetOpen
`  ( Rn `  1o ) ) )  /\  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  X.  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
) ) )  e.  ( Bnd `  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) ) )
8331, 67, 823bitr4d 276 1  |-  ( Y 
C_  RR  ->  ( T  e.  Comp  <->  ( Y  e.  ( Clsd `  U
)  /\  M  e.  ( Bnd `  Y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864    C_ wss 3228   (/)c0 3531   {csn 3716   class class class wbr 4104    e. cmpt 4158    X. cxp 4769   ran crn 4772    |` cres 4773   "cima 4774    o. ccom 4775   -->wf 5333   -1-1-onto->wf1o 5336   ` cfv 5337  (class class class)co 5945   1oc1o 6559    ^m cmap 6860   Fincfn 6951   CCcc 8825   RRcr 8826    - cmin 9127   (,)cioo 10748   abscabs 11815   topGenctg 13441   * Metcxmt 16468   Metcme 16469   MetOpencmopn 16473  TopOnctopon 16738   Clsdccld 16859   Compccmp 17219    Homeo chmeo 17550    ~= chmph 17551   Bndcbnd 25814    Ismty cismty 25845   Rncrrn 25872
This theorem is referenced by:  icccmpALT  25888
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cc 8151  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-omul 6571  df-er 6747  df-ec 6749  df-map 6862  df-pm 6863  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-fi 7255  df-sup 7284  df-oi 7315  df-card 7662  df-acn 7665  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-q 10409  df-rp 10447  df-xneg 10544  df-xadd 10545  df-xmul 10546  df-ioo 10752  df-ico 10754  df-icc 10755  df-fz 10875  df-fzo 10963  df-fl 11017  df-seq 11139  df-exp 11198  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-limsup 12041  df-clim 12058  df-rlim 12059  df-sum 12256  df-gz 13074  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-starv 13320  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-hom 13329  df-cco 13330  df-rest 13426  df-topn 13427  df-topgen 13443  df-prds 13447  df-pws 13449  df-xmet 16475  df-met 16476  df-bl 16477  df-mopn 16478  df-fbas 16479  df-fg 16480  df-cnfld 16483  df-top 16742  df-bases 16744  df-topon 16745  df-topsp 16746  df-cld 16862  df-ntr 16863  df-cls 16864  df-nei 16941  df-cn 17063  df-lm 17065  df-haus 17149  df-cmp 17220  df-hmeo 17552  df-hmph 17553  df-fil 17643  df-fm 17735  df-flim 17736  df-flf 17737  df-xms 17987  df-ms 17988  df-cfil 18785  df-cau 18786  df-cmet 18787  df-totbnd 25815  df-bnd 25826  df-ismty 25846  df-rrn 25873
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