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Theorem imasf1oxms 21504
Description: The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
Hypotheses
Ref Expression
imasf1obl.u  |-  ( ph  ->  U  =  ( F 
"s  R ) )
imasf1obl.v  |-  ( ph  ->  V  =  ( Base `  R ) )
imasf1obl.f  |-  ( ph  ->  F : V -1-1-onto-> B )
imasf1oxms.r  |-  ( ph  ->  R  e.  *MetSp )
Assertion
Ref Expression
imasf1oxms  |-  ( ph  ->  U  e.  *MetSp )

Proof of Theorem imasf1oxms
Dummy variables  x  r  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasf1obl.u . . . . 5  |-  ( ph  ->  U  =  ( F 
"s  R ) )
2 imasf1obl.v . . . . 5  |-  ( ph  ->  V  =  ( Base `  R ) )
3 imasf1obl.f . . . . 5  |-  ( ph  ->  F : V -1-1-onto-> B )
4 imasf1oxms.r . . . . 5  |-  ( ph  ->  R  e.  *MetSp )
5 eqid 2451 . . . . 5  |-  ( (
dist `  R )  |`  ( V  X.  V
) )  =  ( ( dist `  R
)  |`  ( V  X.  V ) )
6 eqid 2451 . . . . 5  |-  ( dist `  U )  =  (
dist `  U )
7 eqid 2451 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
8 eqid 2451 . . . . . . . 8  |-  ( (
dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  =  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) )
97, 8xmsxmet 21471 . . . . . . 7  |-  ( R  e.  *MetSp  ->  (
( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  e.  ( *Met `  ( Base `  R
) ) )
104, 9syl 17 . . . . . 6  |-  ( ph  ->  ( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) )  e.  ( *Met `  ( Base `  R ) ) )
112sqxpeqd 4860 . . . . . . 7  |-  ( ph  ->  ( V  X.  V
)  =  ( (
Base `  R )  X.  ( Base `  R
) ) )
1211reseq2d 5105 . . . . . 6  |-  ( ph  ->  ( ( dist `  R
)  |`  ( V  X.  V ) )  =  ( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) )
132fveq2d 5869 . . . . . 6  |-  ( ph  ->  ( *Met `  V )  =  ( *Met `  ( Base `  R ) ) )
1410, 12, 133eltr4d 2544 . . . . 5  |-  ( ph  ->  ( ( dist `  R
)  |`  ( V  X.  V ) )  e.  ( *Met `  V ) )
151, 2, 3, 4, 5, 6, 14imasf1oxmet 21390 . . . 4  |-  ( ph  ->  ( dist `  U
)  e.  ( *Met `  B ) )
16 f1ofo 5821 . . . . . . 7  |-  ( F : V -1-1-onto-> B  ->  F : V -onto-> B )
173, 16syl 17 . . . . . 6  |-  ( ph  ->  F : V -onto-> B
)
181, 2, 17, 4imasbas 15413 . . . . 5  |-  ( ph  ->  B  =  ( Base `  U ) )
1918fveq2d 5869 . . . 4  |-  ( ph  ->  ( *Met `  B )  =  ( *Met `  ( Base `  U ) ) )
2015, 19eleqtrd 2531 . . 3  |-  ( ph  ->  ( dist `  U
)  e.  ( *Met `  ( Base `  U ) ) )
21 ssid 3451 . . 3  |-  ( Base `  U )  C_  ( Base `  U )
22 xmetres2 21376 . . 3  |-  ( ( ( dist `  U
)  e.  ( *Met `  ( Base `  U ) )  /\  ( Base `  U )  C_  ( Base `  U
) )  ->  (
( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) )  e.  ( *Met `  ( Base `  U
) ) )
2320, 21, 22sylancl 668 . 2  |-  ( ph  ->  ( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) )  e.  ( *Met `  ( Base `  U ) ) )
24 eqid 2451 . . . 4  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
25 eqid 2451 . . . 4  |-  ( TopOpen `  U )  =  (
TopOpen `  U )
261, 2, 17, 4, 24, 25imastopn 20735 . . 3  |-  ( ph  ->  ( TopOpen `  U )  =  ( ( TopOpen `  R ) qTop  F )
)
2724, 7, 8xmstopn 21466 . . . . . 6  |-  ( R  e.  *MetSp  ->  ( TopOpen
`  R )  =  ( MetOpen `  ( ( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) )
284, 27syl 17 . . . . 5  |-  ( ph  ->  ( TopOpen `  R )  =  ( MetOpen `  (
( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) )
2912fveq2d 5869 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  =  ( MetOpen `  ( ( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) )
3028, 29eqtr4d 2488 . . . 4  |-  ( ph  ->  ( TopOpen `  R )  =  ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) ) )
3130oveq1d 6305 . . 3  |-  ( ph  ->  ( ( TopOpen `  R
) qTop  F )  =  ( ( MetOpen `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
) )
32 blbas 21445 . . . . . 6  |-  ( ( ( dist `  R
)  |`  ( V  X.  V ) )  e.  ( *Met `  V )  ->  ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  e.  TopBases )
3314, 32syl 17 . . . . 5  |-  ( ph  ->  ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) )  e.  TopBases )
34 unirnbl 21435 . . . . . . 7  |-  ( ( ( dist `  R
)  |`  ( V  X.  V ) )  e.  ( *Met `  V )  ->  U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  =  V )
35 f1oeq2 5806 . . . . . . 7  |-  ( U. ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) )  =  V  ->  ( F : U. ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) -1-1-onto-> B  <-> 
F : V -1-1-onto-> B ) )
3614, 34, 353syl 18 . . . . . 6  |-  ( ph  ->  ( F : U. ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) -1-1-onto-> B  <->  F : V
-1-1-onto-> B ) )
373, 36mpbird 236 . . . . 5  |-  ( ph  ->  F : U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) -1-1-onto-> B )
38 eqid 2451 . . . . . 6  |-  U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  = 
U. ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )
3938tgqtop 20727 . . . . 5  |-  ( ( ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) )  e.  TopBases 
/\  F : U. ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) -1-1-onto-> B )  ->  ( ( topGen ` 
ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) ) qTop 
F )  =  (
topGen `  ( ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V ) ) ) qTop  F ) ) )
4033, 37, 39syl2anc 667 . . . 4  |-  ( ph  ->  ( ( topGen `  ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) ) qTop 
F )  =  (
topGen `  ( ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V ) ) ) qTop  F ) ) )
41 eqid 2451 . . . . . . 7  |-  ( MetOpen `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )  =  ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) )
4241mopnval 21453 . . . . . 6  |-  ( ( ( dist `  R
)  |`  ( V  X.  V ) )  e.  ( *Met `  V )  ->  ( MetOpen
`  ( ( dist `  R )  |`  ( V  X.  V ) ) )  =  ( topGen ` 
ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) ) )
4314, 42syl 17 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  =  ( topGen `  ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) ) )
4443oveq1d 6305 . . . 4  |-  ( ph  ->  ( ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
)  =  ( (
topGen `  ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) ) qTop  F ) )
45 eqid 2451 . . . . . . 7  |-  ( MetOpen `  ( dist `  U )
)  =  ( MetOpen `  ( dist `  U )
)
4645mopnval 21453 . . . . . 6  |-  ( (
dist `  U )  e.  ( *Met `  B )  ->  ( MetOpen
`  ( dist `  U
) )  =  (
topGen `  ran  ( ball `  ( dist `  U
) ) ) )
4715, 46syl 17 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( dist `  U ) )  =  ( topGen `  ran  ( ball `  ( dist `  U
) ) ) )
48 xmetf 21344 . . . . . . . 8  |-  ( (
dist `  U )  e.  ( *Met `  ( Base `  U )
)  ->  ( dist `  U ) : ( ( Base `  U
)  X.  ( Base `  U ) ) --> RR* )
4920, 48syl 17 . . . . . . 7  |-  ( ph  ->  ( dist `  U
) : ( (
Base `  U )  X.  ( Base `  U
) ) --> RR* )
50 ffn 5728 . . . . . . 7  |-  ( (
dist `  U ) : ( ( Base `  U )  X.  ( Base `  U ) ) -->
RR*  ->  ( dist `  U
)  Fn  ( (
Base `  U )  X.  ( Base `  U
) ) )
51 fnresdm 5685 . . . . . . 7  |-  ( (
dist `  U )  Fn  ( ( Base `  U
)  X.  ( Base `  U ) )  -> 
( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) )  =  ( dist `  U ) )
5249, 50, 513syl 18 . . . . . 6  |-  ( ph  ->  ( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) )  =  ( dist `  U ) )
5352fveq2d 5869 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( ( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) ) )  =  ( MetOpen `  ( dist `  U )
) )
543ad2antrr 732 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  F : V -1-1-onto-> B
)
55 f1of1 5813 . . . . . . . . . . . . . . 15  |-  ( F : V -1-1-onto-> B  ->  F : V -1-1-> B )
5654, 55syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  F : V -1-1-> B )
57 cnvimass 5188 . . . . . . . . . . . . . . 15  |-  ( `' F " x ) 
C_  dom  F
58 f1odm 5818 . . . . . . . . . . . . . . . 16  |-  ( F : V -1-1-onto-> B  ->  dom  F  =  V )
5954, 58syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  dom  F  =  V )
6057, 59syl5sseq 3480 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( `' F " x )  C_  V
)
6114ad2antrr 732 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( ( dist `  R )  |`  ( V  X.  V ) )  e.  ( *Met `  V ) )
62 simprl 764 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  y  e.  V
)
63 simprr 766 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  r  e.  RR* )
64 blssm 21433 . . . . . . . . . . . . . . 15  |-  ( ( ( ( dist `  R
)  |`  ( V  X.  V ) )  e.  ( *Met `  V )  /\  y  e.  V  /\  r  e.  RR* )  ->  (
y ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) r )  C_  V )
6561, 62, 63, 64syl3anc 1268 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( y (
ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) r )  C_  V )
66 f1imaeq 6166 . . . . . . . . . . . . . 14  |-  ( ( F : V -1-1-> B  /\  ( ( `' F " x )  C_  V  /\  ( y ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) r )  C_  V
) )  ->  (
( F " ( `' F " x ) )  =  ( F
" ( y (
ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) r ) )  <->  ( `' F " x )  =  ( y ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) r ) ) )
6756, 60, 65, 66syl12anc 1266 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( ( F
" ( `' F " x ) )  =  ( F " (
y ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) r ) )  <->  ( `' F " x )  =  ( y ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) r ) ) )
6854, 16syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  F : V -onto-> B )
69 simplr 762 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  x  C_  B
)
70 foimacnv 5831 . . . . . . . . . . . . . . 15  |-  ( ( F : V -onto-> B  /\  x  C_  B )  ->  ( F "
( `' F "
x ) )  =  x )
7168, 69, 70syl2anc 667 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( F "
( `' F "
x ) )  =  x )
721ad2antrr 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  U  =  ( F  "s  R ) )
732ad2antrr 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  V  =  (
Base `  R )
)
744ad2antrr 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  R  e.  *MetSp )
7572, 73, 54, 74, 5, 6, 61, 62, 63imasf1obl 21503 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( ( F `
 y ) (
ball `  ( dist `  U ) ) r )  =  ( F
" ( y (
ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) r ) ) )
7675eqcomd 2457 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( F "
( y ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) r ) )  =  ( ( F `  y ) ( ball `  ( dist `  U
) ) r ) )
7771, 76eqeq12d 2466 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( ( F
" ( `' F " x ) )  =  ( F " (
y ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) r ) )  <->  x  =  ( ( F `  y ) ( ball `  ( dist `  U
) ) r ) ) )
7867, 77bitr3d 259 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( ( `' F " x )  =  ( y (
ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) r )  <->  x  =  (
( F `  y
) ( ball `  ( dist `  U ) ) r ) ) )
79782rexbidva 2907 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  B
)  ->  ( E. y  e.  V  E. r  e.  RR*  ( `' F " x )  =  ( y (
ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) r )  <->  E. y  e.  V  E. r  e.  RR*  x  =  ( ( F `
 y ) (
ball `  ( dist `  U ) ) r ) ) )
803adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  B
)  ->  F : V
-1-1-onto-> B )
81 f1ofn 5815 . . . . . . . . . . . 12  |-  ( F : V -1-1-onto-> B  ->  F  Fn  V )
82 oveq1 6297 . . . . . . . . . . . . . . 15  |-  ( z  =  ( F `  y )  ->  (
z ( ball `  ( dist `  U ) ) r )  =  ( ( F `  y
) ( ball `  ( dist `  U ) ) r ) )
8382eqeq2d 2461 . . . . . . . . . . . . . 14  |-  ( z  =  ( F `  y )  ->  (
x  =  ( z ( ball `  ( dist `  U ) ) r )  <->  x  =  ( ( F `  y ) ( ball `  ( dist `  U
) ) r ) ) )
8483rexbidv 2901 . . . . . . . . . . . . 13  |-  ( z  =  ( F `  y )  ->  ( E. r  e.  RR*  x  =  ( z (
ball `  ( dist `  U ) ) r )  <->  E. r  e.  RR*  x  =  ( ( F `  y )
( ball `  ( dist `  U ) ) r ) ) )
8584rexrn 6024 . . . . . . . . . . . 12  |-  ( F  Fn  V  ->  ( E. z  e.  ran  F E. r  e.  RR*  x  =  ( z
( ball `  ( dist `  U ) ) r )  <->  E. y  e.  V  E. r  e.  RR*  x  =  ( ( F `
 y ) (
ball `  ( dist `  U ) ) r ) ) )
8680, 81, 853syl 18 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  B
)  ->  ( E. z  e.  ran  F E. r  e.  RR*  x  =  ( z ( ball `  ( dist `  U
) ) r )  <->  E. y  e.  V  E. r  e.  RR*  x  =  ( ( F `
 y ) (
ball `  ( dist `  U ) ) r ) ) )
87 forn 5796 . . . . . . . . . . . . 13  |-  ( F : V -onto-> B  ->  ran  F  =  B )
8880, 16, 873syl 18 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  B
)  ->  ran  F  =  B )
8988rexeqdv 2994 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  B
)  ->  ( E. z  e.  ran  F E. r  e.  RR*  x  =  ( z ( ball `  ( dist `  U
) ) r )  <->  E. z  e.  B  E. r  e.  RR*  x  =  ( z (
ball `  ( dist `  U ) ) r ) ) )
9079, 86, 893bitr2d 285 . . . . . . . . . 10  |-  ( (
ph  /\  x  C_  B
)  ->  ( E. y  e.  V  E. r  e.  RR*  ( `' F " x )  =  ( y (
ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) r )  <->  E. z  e.  B  E. r  e.  RR*  x  =  ( z (
ball `  ( dist `  U ) ) r ) ) )
9114adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  B
)  ->  ( ( dist `  R )  |`  ( V  X.  V
) )  e.  ( *Met `  V
) )
92 blrn 21424 . . . . . . . . . . 11  |-  ( ( ( dist `  R
)  |`  ( V  X.  V ) )  e.  ( *Met `  V )  ->  (
( `' F "
x )  e.  ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  <->  E. y  e.  V  E. r  e.  RR*  ( `' F " x )  =  ( y ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) r ) ) )
9391, 92syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  x  C_  B
)  ->  ( ( `' F " x )  e.  ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )  <->  E. y  e.  V  E. r  e.  RR*  ( `' F " x )  =  ( y (
ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) r ) ) )
9415adantr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  B
)  ->  ( dist `  U )  e.  ( *Met `  B
) )
95 blrn 21424 . . . . . . . . . . 11  |-  ( (
dist `  U )  e.  ( *Met `  B )  ->  (
x  e.  ran  ( ball `  ( dist `  U
) )  <->  E. z  e.  B  E. r  e.  RR*  x  =  ( z ( ball `  ( dist `  U ) ) r ) ) )
9694, 95syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  x  C_  B
)  ->  ( x  e.  ran  ( ball `  ( dist `  U ) )  <->  E. z  e.  B  E. r  e.  RR*  x  =  ( z (
ball `  ( dist `  U ) ) r ) ) )
9790, 93, 963bitr4d 289 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  B
)  ->  ( ( `' F " x )  e.  ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )  <-> 
x  e.  ran  ( ball `  ( dist `  U
) ) ) )
9897pm5.32da 647 . . . . . . . 8  |-  ( ph  ->  ( ( x  C_  B  /\  ( `' F " x )  e.  ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) )  <-> 
( x  C_  B  /\  x  e.  ran  ( ball `  ( dist `  U ) ) ) ) )
99 f1ofo 5821 . . . . . . . . . 10  |-  ( F : U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V ) ) ) -1-1-onto-> B  ->  F : U. ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) -onto-> B )
10037, 99syl 17 . . . . . . . . 9  |-  ( ph  ->  F : U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) -onto-> B )
10138elqtop2 20716 . . . . . . . . 9  |-  ( ( ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) )  e.  TopBases 
/\  F : U. ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) -onto-> B )  ->  ( x  e.  ( ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) qTop 
F )  <->  ( x  C_  B  /\  ( `' F " x )  e.  ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) ) ) )
10233, 100, 101syl2anc 667 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
)  <->  ( x  C_  B  /\  ( `' F " x )  e.  ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) ) ) )
103 blf 21422 . . . . . . . . . . . 12  |-  ( (
dist `  U )  e.  ( *Met `  B )  ->  ( ball `  ( dist `  U
) ) : ( B  X.  RR* ) --> ~P B )
104 frn 5735 . . . . . . . . . . . 12  |-  ( (
ball `  ( dist `  U ) ) : ( B  X.  RR* )
--> ~P B  ->  ran  ( ball `  ( dist `  U ) )  C_  ~P B )
10515, 103, 1043syl 18 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( ball `  ( dist `  U ) ) 
C_  ~P B )
106105sseld 3431 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ran  ( ball `  ( dist `  U ) )  ->  x  e.  ~P B
) )
107 elpwi 3960 . . . . . . . . . 10  |-  ( x  e.  ~P B  ->  x  C_  B )
108106, 107syl6 34 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ran  ( ball `  ( dist `  U ) )  ->  x  C_  B ) )
109108pm4.71rd 641 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ran  ( ball `  ( dist `  U ) )  <->  ( x  C_  B  /\  x  e. 
ran  ( ball `  ( dist `  U ) ) ) ) )
11098, 102, 1093bitr4d 289 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
)  <->  x  e.  ran  ( ball `  ( dist `  U ) ) ) )
111110eqrdv 2449 . . . . . 6  |-  ( ph  ->  ( ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) qTop 
F )  =  ran  ( ball `  ( dist `  U ) ) )
112111fveq2d 5869 . . . . 5  |-  ( ph  ->  ( topGen `  ( ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
) )  =  (
topGen `  ran  ( ball `  ( dist `  U
) ) ) )
11347, 53, 1123eqtr4d 2495 . . . 4  |-  ( ph  ->  ( MetOpen `  ( ( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) ) )  =  ( topGen `  ( ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) qTop 
F ) ) )
11440, 44, 1133eqtr4d 2495 . . 3  |-  ( ph  ->  ( ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
)  =  ( MetOpen `  ( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) ) ) )
11526, 31, 1143eqtrd 2489 . 2  |-  ( ph  ->  ( TopOpen `  U )  =  ( MetOpen `  (
( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) ) ) )
116 eqid 2451 . . 3  |-  ( Base `  U )  =  (
Base `  U )
117 eqid 2451 . . 3  |-  ( (
dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) )  =  ( ( dist `  U )  |`  (
( Base `  U )  X.  ( Base `  U
) ) )
11825, 116, 117isxms2 21463 . 2  |-  ( U  e.  *MetSp  <->  ( (
( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) )  e.  ( *Met `  ( Base `  U
) )  /\  ( TopOpen
`  U )  =  ( MetOpen `  ( ( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) ) ) ) )
11923, 115, 118sylanbrc 670 1  |-  ( ph  ->  U  e.  *MetSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   E.wrex 2738    C_ wss 3404   ~Pcpw 3951   U.cuni 4198    X. cxp 4832   `'ccnv 4833   dom cdm 4834   ran crn 4835    |` cres 4836   "cima 4837    Fn wfn 5577   -->wf 5578   -1-1->wf1 5579   -onto->wfo 5580   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290   RR*cxr 9674   Basecbs 15121   distcds 15199   TopOpenctopn 15320   topGenctg 15336   qTop cqtop 15401    "s cimas 15402   *Metcxmt 18955   ballcbl 18957   MetOpencmopn 18960   TopBasesctb 19920   *MetSpcxme 21332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  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-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-xrs 15400  df-qtop 15406  df-imas 15407  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-bl 18965  df-mopn 18966  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-xms 21335
This theorem is referenced by:  imasf1oms  21505  xpsxms  21549
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