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Theorem imasf1oxms 20727
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 2467 . . . . 5  |-  ( (
dist `  R )  |`  ( V  X.  V
) )  =  ( ( dist `  R
)  |`  ( V  X.  V ) )
6 eqid 2467 . . . . 5  |-  ( dist `  U )  =  (
dist `  U )
7 eqid 2467 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
8 eqid 2467 . . . . . . . 8  |-  ( (
dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  =  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) )
97, 8xmsxmet 20694 . . . . . . 7  |-  ( R  e.  *MetSp  ->  (
( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  e.  ( *Met `  ( Base `  R
) ) )
104, 9syl 16 . . . . . 6  |-  ( ph  ->  ( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) )  e.  ( *Met `  ( Base `  R ) ) )
112, 2xpeq12d 5024 . . . . . . 7  |-  ( ph  ->  ( V  X.  V
)  =  ( (
Base `  R )  X.  ( Base `  R
) ) )
1211reseq2d 5271 . . . . . 6  |-  ( ph  ->  ( ( dist `  R
)  |`  ( V  X.  V ) )  =  ( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) )
132fveq2d 5868 . . . . . 6  |-  ( ph  ->  ( *Met `  V )  =  ( *Met `  ( Base `  R ) ) )
1410, 12, 133eltr4d 2570 . . . . 5  |-  ( ph  ->  ( ( dist `  R
)  |`  ( V  X.  V ) )  e.  ( *Met `  V ) )
151, 2, 3, 4, 5, 6, 14imasf1oxmet 20613 . . . 4  |-  ( ph  ->  ( dist `  U
)  e.  ( *Met `  B ) )
16 f1ofo 5821 . . . . . . 7  |-  ( F : V -1-1-onto-> B  ->  F : V -onto-> B )
173, 16syl 16 . . . . . 6  |-  ( ph  ->  F : V -onto-> B
)
181, 2, 17, 4imasbas 14763 . . . . 5  |-  ( ph  ->  B  =  ( Base `  U ) )
1918fveq2d 5868 . . . 4  |-  ( ph  ->  ( *Met `  B )  =  ( *Met `  ( Base `  U ) ) )
2015, 19eleqtrd 2557 . . 3  |-  ( ph  ->  ( dist `  U
)  e.  ( *Met `  ( Base `  U ) ) )
21 ssid 3523 . . 3  |-  ( Base `  U )  C_  ( Base `  U )
22 xmetres2 20599 . . 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 662 . 2  |-  ( ph  ->  ( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) )  e.  ( *Met `  ( Base `  U ) ) )
24 eqid 2467 . . . 4  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
25 eqid 2467 . . . 4  |-  ( TopOpen `  U )  =  (
TopOpen `  U )
261, 2, 17, 4, 24, 25imastopn 19956 . . 3  |-  ( ph  ->  ( TopOpen `  U )  =  ( ( TopOpen `  R ) qTop  F )
)
2724, 7, 8xmstopn 20689 . . . . . 6  |-  ( R  e.  *MetSp  ->  ( TopOpen
`  R )  =  ( MetOpen `  ( ( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) )
284, 27syl 16 . . . . 5  |-  ( ph  ->  ( TopOpen `  R )  =  ( MetOpen `  (
( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) )
2912fveq2d 5868 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  =  ( MetOpen `  ( ( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) )
3028, 29eqtr4d 2511 . . . 4  |-  ( ph  ->  ( TopOpen `  R )  =  ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) ) )
3130oveq1d 6297 . . 3  |-  ( ph  ->  ( ( TopOpen `  R
) qTop  F )  =  ( ( MetOpen `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
) )
32 blbas 20668 . . . . . 6  |-  ( ( ( dist `  R
)  |`  ( V  X.  V ) )  e.  ( *Met `  V )  ->  ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  e.  TopBases )
3314, 32syl 16 . . . . 5  |-  ( ph  ->  ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) )  e.  TopBases )
34 unirnbl 20658 . . . . . . 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 20 . . . . . 6  |-  ( ph  ->  ( F : U. ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) -1-1-onto-> B  <->  F : V
-1-1-onto-> B ) )
373, 36mpbird 232 . . . . 5  |-  ( ph  ->  F : U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) -1-1-onto-> B )
38 eqid 2467 . . . . . 6  |-  U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  = 
U. ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )
3938tgqtop 19948 . . . . 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 661 . . . 4  |-  ( ph  ->  ( ( topGen `  ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) ) qTop 
F )  =  (
topGen `  ( ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V ) ) ) qTop  F ) ) )
41 eqid 2467 . . . . . . 7  |-  ( MetOpen `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )  =  ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) )
4241mopnval 20676 . . . . . 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 16 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  =  ( topGen `  ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) ) )
4443oveq1d 6297 . . . 4  |-  ( ph  ->  ( ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
)  =  ( (
topGen `  ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) ) qTop  F ) )
45 eqid 2467 . . . . . . 7  |-  ( MetOpen `  ( dist `  U )
)  =  ( MetOpen `  ( dist `  U )
)
4645mopnval 20676 . . . . . 6  |-  ( (
dist `  U )  e.  ( *Met `  B )  ->  ( MetOpen
`  ( dist `  U
) )  =  (
topGen `  ran  ( ball `  ( dist `  U
) ) ) )
4715, 46syl 16 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( dist `  U ) )  =  ( topGen `  ran  ( ball `  ( dist `  U
) ) ) )
48 xmetf 20567 . . . . . . . 8  |-  ( (
dist `  U )  e.  ( *Met `  ( Base `  U )
)  ->  ( dist `  U ) : ( ( Base `  U
)  X.  ( Base `  U ) ) --> RR* )
4920, 48syl 16 . . . . . . 7  |-  ( ph  ->  ( dist `  U
) : ( (
Base `  U )  X.  ( Base `  U
) ) --> RR* )
50 ffn 5729 . . . . . . 7  |-  ( (
dist `  U ) : ( ( Base `  U )  X.  ( Base `  U ) ) -->
RR*  ->  ( dist `  U
)  Fn  ( (
Base `  U )  X.  ( Base `  U
) ) )
51 fnresdm 5688 . . . . . . 7  |-  ( (
dist `  U )  Fn  ( ( Base `  U
)  X.  ( Base `  U ) )  -> 
( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) )  =  ( dist `  U ) )
5249, 50, 513syl 20 . . . . . 6  |-  ( ph  ->  ( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) )  =  ( dist `  U ) )
5352fveq2d 5868 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( ( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) ) )  =  ( MetOpen `  ( dist `  U )
) )
543ad2antrr 725 . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  F : V -1-1-> B )
57 cnvimass 5355 . . . . . . . . . . . . . . 15  |-  ( `' F " x ) 
C_  dom  F
58 f1odm 5818 . . . . . . . . . . . . . . . 16  |-  ( F : V -1-1-onto-> B  ->  dom  F  =  V )
5954, 58syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  dom  F  =  V )
6057, 59syl5sseq 3552 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( `' F " x )  C_  V
)
6114ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( ( dist `  R )  |`  ( V  X.  V ) )  e.  ( *Met `  V ) )
62 simprl 755 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  y  e.  V
)
63 simprr 756 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  r  e.  RR* )
64 blssm 20656 . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( y (
ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) r )  C_  V )
66 f1imaeq 6159 . . . . . . . . . . . . . 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 1226 . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  F : V -onto-> B )
69 simplr 754 . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( F "
( `' F "
x ) )  =  x )
721ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  U  =  ( F  "s  R ) )
732ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  V  =  (
Base `  R )
)
744ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  R  e.  *MetSp )
7572, 73, 54, 74, 5, 6, 61, 62, 63imasf1obl 20726 . . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . . . 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 2489 . . . . . . . . . . . . 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 255 . . . . . . . . . . . 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 2979 . . . . . . . . . . 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 465 . . . . . . . . . . . 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 6289 . . . . . . . . . . . . . . 15  |-  ( z  =  ( F `  y )  ->  (
z ( ball `  ( dist `  U ) ) r )  =  ( ( F `  y
) ( ball `  ( dist `  U ) ) r ) )
8382eqeq2d 2481 . . . . . . . . . . . . . 14  |-  ( z  =  ( F `  y )  ->  (
x  =  ( z ( ball `  ( dist `  U ) ) r )  <->  x  =  ( ( F `  y ) ( ball `  ( dist `  U
) ) r ) ) )
8483rexbidv 2973 . . . . . . . . . . . . 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 6021 . . . . . . . . . . . 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 20 . . . . . . . . . . 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 20 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  B
)  ->  ran  F  =  B )
8988rexeqdv 3065 . . . . . . . . . . 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 281 . . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  B
)  ->  ( ( dist `  R )  |`  ( V  X.  V
) )  e.  ( *Met `  V
) )
92 blrn 20647 . . . . . . . . . . 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 16 . . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  B
)  ->  ( dist `  U )  e.  ( *Met `  B
) )
95 blrn 20647 . . . . . . . . . . 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 16 . . . . . . . . . 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 285 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  B
)  ->  ( ( `' F " x )  e.  ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )  <-> 
x  e.  ran  ( ball `  ( dist `  U
) ) ) )
9897pm5.32da 641 . . . . . . . 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 16 . . . . . . . . 9  |-  ( ph  ->  F : U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) -onto-> B )
10138elqtop2 19937 . . . . . . . . 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 661 . . . . . . . 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 20645 . . . . . . . . . . . 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 20 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( ball `  ( dist `  U ) ) 
C_  ~P B )
106105sseld 3503 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ran  ( ball `  ( dist `  U ) )  ->  x  e.  ~P B
) )
107 elpwi 4019 . . . . . . . . . 10  |-  ( x  e.  ~P B  ->  x  C_  B )
108106, 107syl6 33 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ran  ( ball `  ( dist `  U ) )  ->  x  C_  B ) )
109108pm4.71rd 635 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ran  ( ball `  ( dist `  U ) )  <->  ( x  C_  B  /\  x  e. 
ran  ( ball `  ( dist `  U ) ) ) ) )
11098, 102, 1093bitr4d 285 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
)  <->  x  e.  ran  ( ball `  ( dist `  U ) ) ) )
111110eqrdv 2464 . . . . . 6  |-  ( ph  ->  ( ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) qTop 
F )  =  ran  ( ball `  ( dist `  U ) ) )
112111fveq2d 5868 . . . . 5  |-  ( ph  ->  ( topGen `  ( ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
) )  =  (
topGen `  ran  ( ball `  ( dist `  U
) ) ) )
11347, 53, 1123eqtr4d 2518 . . . 4  |-  ( ph  ->  ( MetOpen `  ( ( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) ) )  =  ( topGen `  ( ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) qTop 
F ) ) )
11440, 44, 1133eqtr4d 2518 . . 3  |-  ( ph  ->  ( ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
)  =  ( MetOpen `  ( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) ) ) )
11526, 31, 1143eqtrd 2512 . 2  |-  ( ph  ->  ( TopOpen `  U )  =  ( MetOpen `  (
( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) ) ) )
116 eqid 2467 . . 3  |-  ( Base `  U )  =  (
Base `  U )
117 eqid 2467 . . 3  |-  ( (
dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) )  =  ( ( dist `  U )  |`  (
( Base `  U )  X.  ( Base `  U
) ) )
11825, 116, 117isxms2 20686 . 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 664 1  |-  ( ph  ->  U  e.  *MetSp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815    C_ wss 3476   ~Pcpw 4010   U.cuni 4245    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002    Fn wfn 5581   -->wf 5582   -1-1->wf1 5583   -onto->wfo 5584   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   RR*cxr 9623   Basecbs 14486   distcds 14560   TopOpenctopn 14673   topGenctg 14689   qTop cqtop 14754    "s cimas 14755   *Metcxmt 18174   ballcbl 18176   MetOpencmopn 18179   TopBasesctb 19165   *MetSpcxme 20555
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 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-fz 11669  df-fzo 11789  df-seq 12072  df-hash 12370  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-xrs 14753  df-qtop 14758  df-imas 14759  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-bl 18185  df-mopn 18186  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-xms 20558
This theorem is referenced by:  imasf1oms  20728  xpsxms  20772
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