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Theorem reheibor 26563
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 6491 . . . 4  |-  1o  =  { (/) }
2 snfi 6941 . . . 4  |-  { (/) }  e.  Fin
31, 2eqeltri 2353 . . 3  |-  1o  e.  Fin
4 imassrn 5025 . . . . 5  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  C_  ran  ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
5 0ex 4150 . . . . . . . . . 10  |-  (/)  e.  _V
6 eqid 2283 . . . . . . . . . . 11  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
7 eqid 2283 . . . . . . . . . . 11  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  =  ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
86, 7ismrer1 26562 . . . . . . . . . 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 5528 . . . . . . . . . 10  |-  ( Rn
`  1o )  =  ( Rn `  { (/)
} )
1110oveq2i 5869 . . . . . . . . 9  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  1o ) )  =  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Ismty  ( Rn
`  { (/) } ) )
129, 11eleqtrri 2356 . . . . . . . 8  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  e.  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) 
Ismty  ( Rn `  1o ) )
136rexmet 18297 . . . . . . . . 9  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
14 eqid 2283 . . . . . . . . . . 11  |-  ( RR 
^m  1o )  =  ( RR  ^m  1o )
1514rrnmet 26553 . . . . . . . . . 10  |-  ( 1o  e.  Fin  ->  ( Rn `  1o )  e.  ( Met `  ( RR  ^m  1o ) ) )
16 metxmet 17899 . . . . . . . . . 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 26525 . . . . . . . . 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 5472 . . . . . 6  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) ) : RR -1-1-onto-> ( RR  ^m  1o )  ->  ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) : RR --> ( RR  ^m  1o ) )
23 frn 5395 . . . . . 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 3188 . . . 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 2283 . . . 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 2283 . . . 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 2283 . . . 4  |-  ( MetOpen `  ( Rn `  1o ) )  =  ( MetOpen `  ( Rn `  1o ) )
3014, 27, 28, 29rrnheibor 26561 . . 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 18282 . . . . . . . 8  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
34 id 19 . . . . . . . . 9  |-  ( Y 
C_  RR  ->  Y  C_  RR )
35 ax-resscn 8794 . . . . . . . . 9  |-  RR  C_  CC
3634, 35syl6ss 3191 . . . . . . . 8  |-  ( Y 
C_  RR  ->  Y  C_  CC )
37 xmetres2 17925 . . . . . . . 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 2367 . . . . . 6  |-  ( Y 
C_  RR  ->  M  e.  ( * Met `  Y
) )
40 xmetres2 17925 . . . . . . 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 26529 . . . . . 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 2283 . . . . . . . 8  |-  ( ( x  e.  RR  |->  ( { (/) }  X.  {
x } ) )
" Y )  =  ( ( x  e.  RR  |->  ( { (/) }  X.  { x }
) ) " Y
)
49 eqid 2283 . . . . . . . 8  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )  =  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )
5048, 49, 27ismtyres 26532 . . . . . . 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 4792 . . . . . . . . . 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 4984 . . . . . . . . 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 2333 . . . . . . 7  |-  ( Y 
C_  RR  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  ( Y  X.  Y ) )  =  M )
5756oveq1d 5873 . . . . . 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 2359 . . . . 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 3181 . . . 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 17468 . . . 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 17479 . . . 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 17473 . . . . 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 17479 . . . . 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 2283 . . . . . . . . 9  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
706, 69tgioo 18302 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
7168, 70eqtri 2303 . . . . . . 7  |-  U  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) )
7271, 29ismtyhmeo 26529 . . . . . 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 3177 . . . 4  |-  ( x  e.  RR  |->  ( {
(/) }  X.  { x } ) )  e.  ( U  Homeo  ( MetOpen `  ( Rn `  1o ) ) )
75 retopon 18272 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
7668, 75eqeltri 2353 . . . . . 6  |-  U  e.  (TopOn `  RR )
7776toponunii 16670 . . . . 5  |-  RR  =  U. U
7877hmeocld 17458 . . . 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 26531 . . . 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   ran crn 4690    |` cres 4691   "cima 4692    o. ccom 4693   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   1oc1o 6472    ^m cmap 6772   Fincfn 6863   CCcc 8735   RRcr 8736    - cmin 9037   (,)cioo 10656   abscabs 11719   topGenctg 13342   * Metcxmt 16369   Metcme 16370   MetOpencmopn 16372  TopOnctopon 16632   Clsdccld 16753   Compccmp 17113    Homeo chmeo 17444    ~= chmph 17445   Bndcbnd 26491    Ismty cismty 26522   Rncrrn 26549
This theorem is referenced by:  icccmpALT  26565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cc 8061  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-gz 12977  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-prds 13348  df-pws 13350  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-cn 16957  df-lm 16959  df-haus 17043  df-cmp 17114  df-hmeo 17446  df-hmph 17447  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-cfil 18681  df-cau 18682  df-cmet 18683  df-totbnd 26492  df-bnd 26503  df-ismty 26523  df-rrn 26550
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