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Theorem imasf1oxms 18472
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 2404 . . . . 5  |-  ( (
dist `  R )  |`  ( V  X.  V
) )  =  ( ( dist `  R
)  |`  ( V  X.  V ) )
6 eqid 2404 . . . . 5  |-  ( dist `  U )  =  (
dist `  U )
7 eqid 2404 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
8 eqid 2404 . . . . . . . 8  |-  ( (
dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  =  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) )
97, 8xmsxmet 18439 . . . . . . 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 4862 . . . . . . 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 5691 . . . . . 6  |-  ( ph  ->  ( * Met `  V
)  =  ( * Met `  ( Base `  R ) ) )
1410, 12, 133eltr4d 2485 . . . . 5  |-  ( ph  ->  ( ( dist `  R
)  |`  ( V  X.  V ) )  e.  ( * Met `  V
) )
151, 2, 3, 4, 5, 6, 14imasf1oxmet 18358 . . . 4  |-  ( ph  ->  ( dist `  U
)  e.  ( * Met `  B ) )
16 f1ofo 5640 . . . . . . 7  |-  ( F : V -1-1-onto-> B  ->  F : V -onto-> B )
173, 16syl 16 . . . . . 6  |-  ( ph  ->  F : V -onto-> B
)
181, 2, 17, 4imasbas 13693 . . . . 5  |-  ( ph  ->  B  =  ( Base `  U ) )
1918fveq2d 5691 . . . 4  |-  ( ph  ->  ( * Met `  B
)  =  ( * Met `  ( Base `  U ) ) )
2015, 19eleqtrd 2480 . . 3  |-  ( ph  ->  ( dist `  U
)  e.  ( * Met `  ( Base `  U ) ) )
21 ssid 3327 . . 3  |-  ( Base `  U )  C_  ( Base `  U )
22 xmetres2 18344 . . 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 644 . 2  |-  ( ph  ->  ( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) )  e.  ( * Met `  ( Base `  U ) ) )
24 eqid 2404 . . . 4  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
25 eqid 2404 . . . 4  |-  ( TopOpen `  U )  =  (
TopOpen `  U )
261, 2, 17, 4, 24, 25imastopn 17705 . . 3  |-  ( ph  ->  ( TopOpen `  U )  =  ( ( TopOpen `  R ) qTop  F )
)
2724, 7, 8xmstopn 18434 . . . . . 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 5691 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  =  ( MetOpen `  ( ( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) )
3028, 29eqtr4d 2439 . . . 4  |-  ( ph  ->  ( TopOpen `  R )  =  ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) ) )
3130oveq1d 6055 . . 3  |-  ( ph  ->  ( ( TopOpen `  R
) qTop  F )  =  ( ( MetOpen `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
) )
32 blbas 18413 . . . . . 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 18403 . . . . . . 7  |-  ( ( ( dist `  R
)  |`  ( V  X.  V ) )  e.  ( * Met `  V
)  ->  U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V ) ) )  =  V )
35 f1oeq2 5625 . . . . . . 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 19 . . . . . 6  |-  ( ph  ->  ( F : U. ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) -1-1-onto-> B  <->  F : V
-1-1-onto-> B ) )
373, 36mpbird 224 . . . . 5  |-  ( ph  ->  F : U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) -1-1-onto-> B )
38 eqid 2404 . . . . . 6  |-  U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  = 
U. ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )
3938tgqtop 17697 . . . . 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 643 . . . 4  |-  ( ph  ->  ( ( topGen `  ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) ) qTop 
F )  =  (
topGen `  ( ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V ) ) ) qTop  F ) ) )
41 eqid 2404 . . . . . . 7  |-  ( MetOpen `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )  =  ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) )
4241mopnval 18421 . . . . . 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 6055 . . . 4  |-  ( ph  ->  ( ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
)  =  ( (
topGen `  ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) ) qTop  F ) )
45 eqid 2404 . . . . . . 7  |-  ( MetOpen `  ( dist `  U )
)  =  ( MetOpen `  ( dist `  U )
)
4645mopnval 18421 . . . . . 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 18312 . . . . . . . 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 5550 . . . . . . 7  |-  ( (
dist `  U ) : ( ( Base `  U )  X.  ( Base `  U ) ) -->
RR*  ->  ( dist `  U
)  Fn  ( (
Base `  U )  X.  ( Base `  U
) ) )
51 fnresdm 5513 . . . . . . 7  |-  ( (
dist `  U )  Fn  ( ( Base `  U
)  X.  ( Base `  U ) )  -> 
( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) )  =  ( dist `  U ) )
5249, 50, 513syl 19 . . . . . 6  |-  ( ph  ->  ( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) )  =  ( dist `  U ) )
5352fveq2d 5691 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( ( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) ) )  =  ( MetOpen `  ( dist `  U )
) )
543ad2antrr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  F : V -1-1-onto-> B
)
55 f1of1 5632 . . . . . . . . . . . . . . 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 5183 . . . . . . . . . . . . . . 15  |-  ( `' F " x ) 
C_  dom  F
58 f1odm 5637 . . . . . . . . . . . . . . . 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 3356 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( `' F " x )  C_  V
)
6114ad2antrr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( ( dist `  R )  |`  ( V  X.  V ) )  e.  ( * Met `  V ) )
62 simprl 733 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  y  e.  V
)
63 simprr 734 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  r  e.  RR* )
64 blssm 18401 . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( y (
ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) r )  C_  V )
66 f1imaeq 5970 . . . . . . . . . . . . . 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 1182 . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  x  C_  B
)
70 foimacnv 5651 . . . . . . . . . . . . . . 15  |-  ( ( F : V -onto-> B  /\  x  C_  B )  ->  ( F "
( `' F "
x ) )  =  x )
7168, 69, 70syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( F "
( `' F "
x ) )  =  x )
721ad2antrr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  U  =  ( F  "s  R ) )
732ad2antrr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  V  =  (
Base `  R )
)
744ad2antrr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  R  e.  * MetSp )
7572, 73, 54, 74, 5, 6, 61, 62, 63imasf1obl 18471 . . . . . . . . . . . . . . 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 2409 . . . . . . . . . . . . . 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 2418 . . . . . . . . . . . . 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 247 . . . . . . . . . . . 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 2707 . . . . . . . . . . 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 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  B
)  ->  F : V
-1-1-onto-> B )
81 f1ofn 5634 . . . . . . . . . . . 12  |-  ( F : V -1-1-onto-> B  ->  F  Fn  V )
82 oveq1 6047 . . . . . . . . . . . . . . 15  |-  ( z  =  ( F `  y )  ->  (
z ( ball `  ( dist `  U ) ) r )  =  ( ( F `  y
) ( ball `  ( dist `  U ) ) r ) )
8382eqeq2d 2415 . . . . . . . . . . . . . 14  |-  ( z  =  ( F `  y )  ->  (
x  =  ( z ( ball `  ( dist `  U ) ) r )  <->  x  =  ( ( F `  y ) ( ball `  ( dist `  U
) ) r ) ) )
8483rexbidv 2687 . . . . . . . . . . . . 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 5831 . . . . . . . . . . . 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 19 . . . . . . . . . . 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 5615 . . . . . . . . . . . . 13  |-  ( F : V -onto-> B  ->  ran  F  =  B )
8880, 16, 873syl 19 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  B
)  ->  ran  F  =  B )
8988rexeqdv 2871 . . . . . . . . . . 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 273 . . . . . . . . . 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 452 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  B
)  ->  ( ( dist `  R )  |`  ( V  X.  V
) )  e.  ( * Met `  V
) )
92 blrn 18392 . . . . . . . . . . 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 452 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  B
)  ->  ( dist `  U )  e.  ( * Met `  B
) )
95 blrn 18392 . . . . . . . . . . 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 277 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  B
)  ->  ( ( `' F " x )  e.  ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )  <-> 
x  e.  ran  ( ball `  ( dist `  U
) ) ) )
9897pm5.32da 623 . . . . . . . 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 5640 . . . . . . . . . 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 17686 . . . . . . . . 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 643 . . . . . . . 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 18390 . . . . . . . . . . . 12  |-  ( (
dist `  U )  e.  ( * Met `  B
)  ->  ( ball `  ( dist `  U
) ) : ( B  X.  RR* ) --> ~P B )
104 frn 5556 . . . . . . . . . . . 12  |-  ( (
ball `  ( dist `  U ) ) : ( B  X.  RR* )
--> ~P B  ->  ran  ( ball `  ( dist `  U ) )  C_  ~P B )
10515, 103, 1043syl 19 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( ball `  ( dist `  U ) ) 
C_  ~P B )
106105sseld 3307 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ran  ( ball `  ( dist `  U ) )  ->  x  e.  ~P B
) )
107 elpwi 3767 . . . . . . . . . 10  |-  ( x  e.  ~P B  ->  x  C_  B )
108106, 107syl6 31 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ran  ( ball `  ( dist `  U ) )  ->  x  C_  B ) )
109108pm4.71rd 617 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ran  ( ball `  ( dist `  U ) )  <->  ( x  C_  B  /\  x  e. 
ran  ( ball `  ( dist `  U ) ) ) ) )
11098, 102, 1093bitr4d 277 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
)  <->  x  e.  ran  ( ball `  ( dist `  U ) ) ) )
111110eqrdv 2402 . . . . . 6  |-  ( ph  ->  ( ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) qTop 
F )  =  ran  ( ball `  ( dist `  U ) ) )
112111fveq2d 5691 . . . . 5  |-  ( ph  ->  ( topGen `  ( ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
) )  =  (
topGen `  ran  ( ball `  ( dist `  U
) ) ) )
11347, 53, 1123eqtr4d 2446 . . . 4  |-  ( ph  ->  ( MetOpen `  ( ( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) ) )  =  ( topGen `  ( ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) qTop 
F ) ) )
11440, 44, 1133eqtr4d 2446 . . 3  |-  ( ph  ->  ( ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
)  =  ( MetOpen `  ( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) ) ) )
11526, 31, 1143eqtrd 2440 . 2  |-  ( ph  ->  ( TopOpen `  U )  =  ( MetOpen `  (
( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) ) ) )
116 eqid 2404 . . 3  |-  ( Base `  U )  =  (
Base `  U )
117 eqid 2404 . . 3  |-  ( (
dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) )  =  ( ( dist `  U )  |`  (
( Base `  U )  X.  ( Base `  U
) ) )
11825, 116, 117isxms2 18431 . 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 646 1  |-  ( ph  ->  U  e.  * MetSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667    C_ wss 3280   ~Pcpw 3759   U.cuni 3975    X. cxp 4835   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840    Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   -onto->wfo 5411   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   RR*cxr 9075   Basecbs 13424   distcds 13493   TopOpenctopn 13604   topGenctg 13620   qTop cqtop 13684    "s cimas 13685   * Metcxmt 16641   ballcbl 16643   MetOpencmopn 16646   TopBasesctb 16917   * MetSpcxme 18300
This theorem is referenced by:  imasf1oms  18473  xpsxms  18517
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-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
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-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  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-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  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-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-rest 13605  df-topn 13606  df-topgen 13622  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-xms 18303
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