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Theorem reheibor 31875
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 7202 . . . 4  |-  1o  =  { (/) }
2 snfi 7657 . . . 4  |-  { (/) }  e.  Fin
31, 2eqeltri 2513 . . 3  |-  1o  e.  Fin
4 imassrn 5199 . . . . 5  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  C_  ran  ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
5 0ex 4557 . . . . . . . . . 10  |-  (/)  e.  _V
6 eqid 2429 . . . . . . . . . . 11  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
7 eqid 2429 . . . . . . . . . . 11  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  =  ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
86, 7ismrer1 31874 . . . . . . . . . 10  |-  ( (/)  e.  _V  ->  ( x  e.  RR  |->  ( { (/) }  X.  { x }
) )  e.  ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) 
Ismty  ( Rn `  { (/)
} ) ) )
95, 8ax-mp 5 . . . . . . . . 9  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  e.  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) 
Ismty  ( Rn `  { (/)
} ) )
101fveq2i 5884 . . . . . . . . . 10  |-  ( Rn
`  1o )  =  ( Rn `  { (/)
} )
1110oveq2i 6316 . . . . . . . . 9  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  1o ) )  =  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  { (/) } ) )
129, 11eleqtrri 2516 . . . . . . . 8  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  e.  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) 
Ismty  ( Rn `  1o ) )
136rexmet 21720 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR )
14 eqid 2429 . . . . . . . . . . 11  |-  ( RR 
^m  1o )  =  ( RR  ^m  1o )
1514rrnmet 31865 . . . . . . . . . 10  |-  ( 1o  e.  Fin  ->  ( Rn `  1o )  e.  ( Met `  ( RR  ^m  1o ) ) )
16 metxmet 21280 . . . . . . . . . 10  |-  ( ( Rn `  1o )  e.  ( Met `  ( RR  ^m  1o ) )  ->  ( Rn `  1o )  e.  ( *Met `  ( RR 
^m  1o ) ) )
173, 15, 16mp2b 10 . . . . . . . . 9  |-  ( Rn
`  1o )  e.  ( *Met `  ( RR  ^m  1o ) )
18 isismty 31837 . . . . . . . . 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 676 . . . . . . . 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 211 . . . . . . 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 459 . . . . . 6  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) : RR -1-1-onto-> ( RR  ^m  1o )
22 f1of 5831 . . . . . 6  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) ) : RR -1-1-onto-> ( RR  ^m  1o )  ->  ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) : RR --> ( RR  ^m  1o ) )
23 frn 5752 . . . . . 6  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) ) : RR --> ( RR 
^m  1o )  ->  ran  ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) ) 
C_  ( RR  ^m  1o ) )
2421, 22, 23mp2b 10 . . . . 5  |-  ran  (
x  e.  RR  |->  ( { (/) }  X.  {
x } ) ) 
C_  ( RR  ^m  1o )
254, 24sstri 3479 . . . 4  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  C_  ( RR  ^m  1o )
2625a1i 11 . . 3  |-  ( Y 
C_  RR  ->  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  C_  ( RR  ^m  1o ) )
27 eqid 2429 . . . 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 2429 . . . 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 2429 . . . 4  |-  ( MetOpen `  ( Rn `  1o ) )  =  ( MetOpen `  ( Rn `  1o ) )
3014, 27, 28, 29rrnheibor 31873 . . 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 667 . 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 21704 . . . . . . . 8  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
34 id 23 . . . . . . . . 9  |-  ( Y 
C_  RR  ->  Y  C_  RR )
35 ax-resscn 9595 . . . . . . . . 9  |-  RR  C_  CC
3634, 35syl6ss 3482 . . . . . . . 8  |-  ( Y 
C_  RR  ->  Y  C_  CC )
37 xmetres2 21307 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  Y  C_  CC )  -> 
( ( abs  o.  -  )  |`  ( Y  X.  Y ) )  e.  ( *Met `  Y ) )
3833, 36, 37sylancr 667 . . . . . . 7  |-  ( Y 
C_  RR  ->  ( ( abs  o.  -  )  |`  ( Y  X.  Y
) )  e.  ( *Met `  Y
) )
3932, 38syl5eqel 2521 . . . . . 6  |-  ( Y 
C_  RR  ->  M  e.  ( *Met `  Y ) )
40 xmetres2 21307 . . . . . . 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 667 . . . . . 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 31841 . . . . . 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 665 . . . . 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 11 . . . . . . 7  |-  ( Y 
C_  RR  ->  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR ) )
4617a1i 11 . . . . . . 7  |-  ( Y 
C_  RR  ->  ( Rn
`  1o )  e.  ( *Met `  ( RR  ^m  1o ) ) )
4712a1i 11 . . . . . . 7  |-  ( Y 
C_  RR  ->  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  e.  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) 
Ismty  ( Rn `  1o ) ) )
48 eqid 2429 . . . . . . . 8  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  =  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
)
49 eqid 2429 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )  =  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )
5048, 49, 27ismtyres 31844 . . . . . . 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 1265 . . . . . 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 4960 . . . . . . . . . 10  |-  ( ( Y  C_  RR  /\  Y  C_  RR )  ->  ( Y  X.  Y )  C_  ( RR  X.  RR ) )
5352anidms 649 . . . . . . . . 9  |-  ( Y 
C_  RR  ->  ( Y  X.  Y )  C_  ( RR  X.  RR ) )
5453resabs1d 5154 . . . . . . . 8  |-  ( Y 
C_  RR  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )  =  ( ( abs 
o.  -  )  |`  ( Y  X.  Y ) ) )
5554, 32syl6eqr 2488 . . . . . . 7  |-  ( Y 
C_  RR  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )  =  M )
5655oveq1d 6320 . . . . . 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 ) ) ) ) )
5751, 56eleqtrd 2519 . . . . 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 ) ) ) ) )
5844, 57sseldd 3471 . . . 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
) ) ) ) ) )
59 hmphi 20723 . . . 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 ) ) ) ) )
6058, 59syl 17 . . 3  |-  ( Y 
C_  RR  ->  T  ~=  ( MetOpen `  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) ) )
61 cmphmph 20734 . . . 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 ) )
62 hmphsym 20728 . . . . 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 )
63 cmphmph 20734 . . . . 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 ) )
6462, 63syl 17 . . . 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 ) )
6561, 64impbid 193 . . 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 ) )
6660, 65syl 17 . 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 ) )
67 reheibor.4 . . . . . . . 8  |-  U  =  ( topGen `  ran  (,) )
68 eqid 2429 . . . . . . . . 9  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
696, 68tgioo 21725 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
7067, 69eqtri 2458 . . . . . . 7  |-  U  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
7170, 29ismtyhmeo 31841 . . . . . 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 ) ) ) )
7213, 17, 71mp2an 676 . . . . 5  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  1o ) ) 
C_  ( U Homeo (
MetOpen `  ( Rn `  1o ) ) )
7372, 12sselii 3467 . . . 4  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  e.  ( U Homeo ( MetOpen `  ( Rn `  1o ) ) )
74 retopon 21695 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
7567, 74eqeltri 2513 . . . . . 6  |-  U  e.  (TopOn `  RR )
7675toponunii 19878 . . . . 5  |-  RR  =  U. U
7776hmeocld 20713 . . . 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 ) ) ) ) )
7873, 34, 77sylancr 667 . . 3  |-  ( Y 
C_  RR  ->  ( Y  e.  ( Clsd `  U
)  <->  ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  e.  (
Clsd `  ( MetOpen `  ( Rn `  1o ) ) ) ) )
79 ismtybnd 31843 . . . 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 ) ) ) )
8039, 41, 57, 79syl3anc 1264 . . 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
) ) ) )
8178, 80anbi12d 715 . 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 ) ) ) ) )
8231, 66, 813bitr4d 288 1  |-  ( Y 
C_  RR  ->  ( T  e.  Comp  <->  ( Y  e.  ( Clsd `  U
)  /\  M  e.  ( Bnd `  Y ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   _Vcvv 3087    C_ wss 3442   (/)c0 3767   {csn 4002   class class class wbr 4426    |-> cmpt 4484    X. cxp 4852   ran crn 4855    |` cres 4856   "cima 4857    o. ccom 4858   -->wf 5597   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305   1oc1o 7183    ^m cmap 7480   Fincfn 7577   CCcc 9536   RRcr 9537    - cmin 9859   (,)cioo 11635   abscabs 13276   topGenctg 15295   *Metcxmt 18890   Metcme 18891   MetOpencmopn 18895  TopOnctopon 19849   Clsdccld 19962   Compccmp 20332   Homeochmeo 20699    ~= chmph 20700   Bndcbnd 31803    Ismty cismty 31834   Rncrrn 31861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cc 8863  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-omul 7195  df-er 7371  df-ec 7373  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fi 7931  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-acn 8375  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-sum 13731  df-gz 14837  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-topgen 15301  df-prds 15305  df-pws 15307  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-cn 20174  df-lm 20176  df-haus 20262  df-cmp 20333  df-hmeo 20701  df-hmph 20702  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-xms 21266  df-ms 21267  df-cfil 22118  df-cau 22119  df-cmet 22120  df-totbnd 31804  df-bnd 31815  df-ismty 31835  df-rrn 31862
This theorem is referenced by:  icccmpALT  31877
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