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Theorem reheibor 26438
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 6695 . . . 4  |-  1o  =  { (/) }
2 snfi 7146 . . . 4  |-  { (/) }  e.  Fin
31, 2eqeltri 2474 . . 3  |-  1o  e.  Fin
4 imassrn 5175 . . . . 5  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  C_  ran  ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
5 0ex 4299 . . . . . . . . . 10  |-  (/)  e.  _V
6 eqid 2404 . . . . . . . . . . 11  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
7 eqid 2404 . . . . . . . . . . 11  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  =  ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
86, 7ismrer1 26437 . . . . . . . . . 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 5690 . . . . . . . . . 10  |-  ( Rn
`  1o )  =  ( Rn `  { (/)
} )
1110oveq2i 6051 . . . . . . . . 9  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  1o ) )  =  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  { (/) } ) )
129, 11eleqtrri 2477 . . . . . . . 8  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  e.  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) 
Ismty  ( Rn `  1o ) )
136rexmet 18775 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
14 eqid 2404 . . . . . . . . . . 11  |-  ( RR 
^m  1o )  =  ( RR  ^m  1o )
1514rrnmet 26428 . . . . . . . . . 10  |-  ( 1o  e.  Fin  ->  ( Rn `  1o )  e.  ( Met `  ( RR  ^m  1o ) ) )
16 metxmet 18317 . . . . . . . . . 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 26400 . . . . . . . . 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 654 . . . . . . . 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 200 . . . . . . 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 445 . . . . . 6  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) : RR -1-1-onto-> ( RR  ^m  1o )
22 f1of 5633 . . . . . 6  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) ) : RR -1-1-onto-> ( RR  ^m  1o )  ->  ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) : RR --> ( RR  ^m  1o ) )
23 frn 5556 . . . . . 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 3317 . . . 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 2404 . . . 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 2404 . . . 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 2404 . . . 4  |-  ( MetOpen `  ( Rn `  1o ) )  =  ( MetOpen `  ( Rn `  1o ) )
3014, 27, 28, 29rrnheibor 26436 . . 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 645 . 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 18760 . . . . . . . 8  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
34 id 20 . . . . . . . . 9  |-  ( Y 
C_  RR  ->  Y  C_  RR )
35 ax-resscn 9003 . . . . . . . . 9  |-  RR  C_  CC
3634, 35syl6ss 3320 . . . . . . . 8  |-  ( Y 
C_  RR  ->  Y  C_  CC )
37 xmetres2 18344 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  e.  ( * Met `  CC )  /\  Y  C_  CC )  -> 
( ( abs  o.  -  )  |`  ( Y  X.  Y ) )  e.  ( * Met `  Y ) )
3833, 36, 37sylancr 645 . . . . . . 7  |-  ( Y 
C_  RR  ->  ( ( abs  o.  -  )  |`  ( Y  X.  Y
) )  e.  ( * Met `  Y
) )
3932, 38syl5eqel 2488 . . . . . 6  |-  ( Y 
C_  RR  ->  M  e.  ( * Met `  Y
) )
40 xmetres2 18344 . . . . . . 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 645 . . . . . 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 26404 . . . . . 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 643 . . . . 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 2404 . . . . . . . 8  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  =  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
)
49 eqid 2404 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )  =  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )
5048, 49, 27ismtyres 26407 . . . . . . 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 1185 . . . . . 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 4940 . . . . . . . . . 10  |-  ( ( Y  C_  RR  /\  Y  C_  RR )  ->  ( Y  X.  Y )  C_  ( RR  X.  RR ) )
5352anidms 627 . . . . . . . . 9  |-  ( Y 
C_  RR  ->  ( Y  X.  Y )  C_  ( RR  X.  RR ) )
54 resabs1 5134 . . . . . . . . 9  |-  ( ( Y  X.  Y ) 
C_  ( RR  X.  RR )  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )  =  ( ( abs 
o.  -  )  |`  ( Y  X.  Y ) ) )
5553, 54syl 16 . . . . . . . 8  |-  ( Y 
C_  RR  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )  =  ( ( abs 
o.  -  )  |`  ( Y  X.  Y ) ) )
5655, 32syl6eqr 2454 . . . . . . 7  |-  ( Y 
C_  RR  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )  =  M )
5756oveq1d 6055 . . . . . 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 2480 . . . . 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 3309 . . . 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 17762 . . . 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 16 . . 3  |-  ( Y 
C_  RR  ->  T  ~=  ( MetOpen `  ( ( Rn `  1o )  |`  ( ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  X.  (
( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y ) ) ) ) )
62 cmphmph 17773 . . . 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 17767 . . . . 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 17773 . . . . 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 16 . . . 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 184 . . 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 16 . 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 2404 . . . . . . . . 9  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
706, 69tgioo 18780 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
7168, 70eqtri 2424 . . . . . . 7  |-  U  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
7271, 29ismtyhmeo 26404 . . . . . 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 654 . . . . 5  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  1o ) ) 
C_  ( U  Homeo  (
MetOpen `  ( Rn `  1o ) ) )
7473, 12sselii 3305 . . . 4  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  e.  ( U  Homeo  ( MetOpen `  ( Rn `  1o ) ) )
75 retopon 18750 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
7668, 75eqeltri 2474 . . . . . 6  |-  U  e.  (TopOn `  RR )
7776toponunii 16952 . . . . 5  |-  RR  =  U. U
7877hmeocld 17752 . . . 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 645 . . 3  |-  ( Y 
C_  RR  ->  ( Y  e.  ( Clsd `  U
)  <->  ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  e.  (
Clsd `  ( MetOpen `  ( Rn `  1o ) ) ) ) )
80 ismtybnd 26406 . . . 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 1184 . . 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 692 . 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 277 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    C_ wss 3280   (/)c0 3588   {csn 3774   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   ran crn 4838    |` cres 4839   "cima 4840    o. ccom 4841   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   1oc1o 6676    ^m cmap 6977   Fincfn 7068   CCcc 8944   RRcr 8945    - cmin 9247   (,)cioo 10872   abscabs 11994   topGenctg 13620   * Metcxmt 16641   Metcme 16642   MetOpencmopn 16646  TopOnctopon 16914   Clsdccld 17035   Compccmp 17403    Homeo chmeo 17738    ~= chmph 17739   Bndcbnd 26366    Ismty cismty 26397   Rncrrn 26424
This theorem is referenced by:  icccmpALT  26440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cc 8271  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-er 6864  df-ec 6866  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-gz 13253  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-prds 13626  df-pws 13628  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-cn 17245  df-lm 17247  df-haus 17333  df-cmp 17404  df-hmeo 17740  df-hmph 17741  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-cfil 19161  df-cau 19162  df-cmet 19163  df-totbnd 26367  df-bnd 26378  df-ismty 26398  df-rrn 26425
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