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Theorem reheibor 29966
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 7142 . . . 4  |-  1o  =  { (/) }
2 snfi 7596 . . . 4  |-  { (/) }  e.  Fin
31, 2eqeltri 2551 . . 3  |-  1o  e.  Fin
4 imassrn 5348 . . . . 5  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  C_  ran  ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
5 0ex 4577 . . . . . . . . . 10  |-  (/)  e.  _V
6 eqid 2467 . . . . . . . . . . 11  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
7 eqid 2467 . . . . . . . . . . 11  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  =  ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
86, 7ismrer1 29965 . . . . . . . . . 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 5869 . . . . . . . . . 10  |-  ( Rn
`  1o )  =  ( Rn `  { (/)
} )
1110oveq2i 6295 . . . . . . . . 9  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  1o ) )  =  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  { (/) } ) )
129, 11eleqtrri 2554 . . . . . . . 8  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  e.  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) 
Ismty  ( Rn `  1o ) )
136rexmet 21059 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR )
14 eqid 2467 . . . . . . . . . . 11  |-  ( RR 
^m  1o )  =  ( RR  ^m  1o )
1514rrnmet 29956 . . . . . . . . . 10  |-  ( 1o  e.  Fin  ->  ( Rn `  1o )  e.  ( Met `  ( RR  ^m  1o ) ) )
16 metxmet 20600 . . . . . . . . . 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 29928 . . . . . . . . 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 672 . . . . . . . 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 208 . . . . . . 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 458 . . . . . 6  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) : RR -1-1-onto-> ( RR  ^m  1o )
22 f1of 5816 . . . . . 6  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) ) : RR -1-1-onto-> ( RR  ^m  1o )  ->  ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) : RR --> ( RR  ^m  1o ) )
23 frn 5737 . . . . . 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 3513 . . . 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 2467 . . . 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 2467 . . . 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 2467 . . . 4  |-  ( MetOpen `  ( Rn `  1o ) )  =  ( MetOpen `  ( Rn `  1o ) )
3014, 27, 28, 29rrnheibor 29964 . . 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 663 . 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 21043 . . . . . . . 8  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
34 id 22 . . . . . . . . 9  |-  ( Y 
C_  RR  ->  Y  C_  RR )
35 ax-resscn 9549 . . . . . . . . 9  |-  RR  C_  CC
3634, 35syl6ss 3516 . . . . . . . 8  |-  ( Y 
C_  RR  ->  Y  C_  CC )
37 xmetres2 20627 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  Y  C_  CC )  -> 
( ( abs  o.  -  )  |`  ( Y  X.  Y ) )  e.  ( *Met `  Y ) )
3833, 36, 37sylancr 663 . . . . . . 7  |-  ( Y 
C_  RR  ->  ( ( abs  o.  -  )  |`  ( Y  X.  Y
) )  e.  ( *Met `  Y
) )
3932, 38syl5eqel 2559 . . . . . 6  |-  ( Y 
C_  RR  ->  M  e.  ( *Met `  Y ) )
40 xmetres2 20627 . . . . . . 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 663 . . . . . 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 29932 . . . . . 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 661 . . . . 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 2467 . . . . . . . 8  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  =  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
)
49 eqid 2467 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )  =  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )
5048, 49, 27ismtyres 29935 . . . . . . 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 1229 . . . . . 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 5108 . . . . . . . . . 10  |-  ( ( Y  C_  RR  /\  Y  C_  RR )  ->  ( Y  X.  Y )  C_  ( RR  X.  RR ) )
5352anidms 645 . . . . . . . . 9  |-  ( Y 
C_  RR  ->  ( Y  X.  Y )  C_  ( RR  X.  RR ) )
54 resabs1 5302 . . . . . . . . 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 2526 . . . . . . 7  |-  ( Y 
C_  RR  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )  =  M )
5756oveq1d 6299 . . . . . 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 2557 . . . . 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 3505 . . . 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 20041 . . . 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 20052 . . . 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 20046 . . . . 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 20052 . . . . 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 191 . . 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 2467 . . . . . . . . 9  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
706, 69tgioo 21064 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
7168, 70eqtri 2496 . . . . . . 7  |-  U  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
7271, 29ismtyhmeo 29932 . . . . . 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 672 . . . . 5  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  1o ) ) 
C_  ( U Homeo (
MetOpen `  ( Rn `  1o ) ) )
7473, 12sselii 3501 . . . 4  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  e.  ( U Homeo ( MetOpen `  ( Rn `  1o ) ) )
75 retopon 21033 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
7668, 75eqeltri 2551 . . . . . 6  |-  U  e.  (TopOn `  RR )
7776toponunii 19228 . . . . 5  |-  RR  =  U. U
7877hmeocld 20031 . . . 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 663 . . 3  |-  ( Y 
C_  RR  ->  ( Y  e.  ( Clsd `  U
)  <->  ( ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) ) " Y )  e.  (
Clsd `  ( MetOpen `  ( Rn `  1o ) ) ) ) )
80 ismtybnd 29934 . . . 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 1228 . . 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 710 . 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 285 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 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    C_ wss 3476   (/)c0 3785   {csn 4027   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   ran crn 5000    |` cres 5001   "cima 5002    o. ccom 5003   -->wf 5584   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284   1oc1o 7123    ^m cmap 7420   Fincfn 7516   CCcc 9490   RRcr 9491    - cmin 9805   (,)cioo 11529   abscabs 13030   topGenctg 14693   *Metcxmt 18202   Metcme 18203   MetOpencmopn 18207  TopOnctopon 19190   Clsdccld 19311   Compccmp 19680   Homeochmeo 20017    ~= chmph 20018   Bndcbnd 29894    Ismty cismty 29925   Rncrrn 29952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cc 8815  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-omul 7135  df-er 7311  df-ec 7313  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-acn 8323  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-limsup 13257  df-clim 13274  df-rlim 13275  df-sum 13472  df-gz 14307  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-topgen 14699  df-prds 14703  df-pws 14705  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-cn 19522  df-lm 19524  df-haus 19610  df-cmp 19681  df-hmeo 20019  df-hmph 20020  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-xms 20586  df-ms 20587  df-cfil 21457  df-cau 21458  df-cmet 21459  df-totbnd 29895  df-bnd 29906  df-ismty 29926  df-rrn 29953
This theorem is referenced by:  icccmpALT  29968
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