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Theorem imasf1oxms 18035
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 2283 . . . . 5  |-  ( (
dist `  R )  |`  ( V  X.  V
) )  =  ( ( dist `  R
)  |`  ( V  X.  V ) )
6 eqid 2283 . . . . 5  |-  ( dist `  U )  =  (
dist `  U )
7 eqid 2283 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
8 eqid 2283 . . . . . . . 8  |-  ( (
dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  =  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) )
97, 8xmsxmet 18002 . . . . . . 7  |-  ( R  e.  * MetSp  ->  (
( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  e.  ( * Met `  ( Base `  R
) ) )
104, 9syl 15 . . . . . 6  |-  ( ph  ->  ( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) )  e.  ( * Met `  ( Base `  R ) ) )
112, 2xpeq12d 4714 . . . . . . . 8  |-  ( ph  ->  ( V  X.  V
)  =  ( (
Base `  R )  X.  ( Base `  R
) ) )
1211reseq2d 4955 . . . . . . 7  |-  ( ph  ->  ( ( dist `  R
)  |`  ( V  X.  V ) )  =  ( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) )
132fveq2d 5529 . . . . . . 7  |-  ( ph  ->  ( * Met `  V
)  =  ( * Met `  ( Base `  R ) ) )
1412, 13eleq12d 2351 . . . . . 6  |-  ( ph  ->  ( ( ( dist `  R )  |`  ( V  X.  V ) )  e.  ( * Met `  V )  <->  ( ( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  e.  ( * Met `  ( Base `  R
) ) ) )
1510, 14mpbird 223 . . . . 5  |-  ( ph  ->  ( ( dist `  R
)  |`  ( V  X.  V ) )  e.  ( * Met `  V
) )
161, 2, 3, 4, 5, 6, 15imasf1oxmet 17939 . . . 4  |-  ( ph  ->  ( dist `  U
)  e.  ( * Met `  B ) )
17 f1ofo 5479 . . . . . . 7  |-  ( F : V -1-1-onto-> B  ->  F : V -onto-> B )
183, 17syl 15 . . . . . 6  |-  ( ph  ->  F : V -onto-> B
)
191, 2, 18, 4imasbas 13415 . . . . 5  |-  ( ph  ->  B  =  ( Base `  U ) )
2019fveq2d 5529 . . . 4  |-  ( ph  ->  ( * Met `  B
)  =  ( * Met `  ( Base `  U ) ) )
2116, 20eleqtrd 2359 . . 3  |-  ( ph  ->  ( dist `  U
)  e.  ( * Met `  ( Base `  U ) ) )
22 ssid 3197 . . 3  |-  ( Base `  U )  C_  ( Base `  U )
23 xmetres2 17925 . . 3  |-  ( ( ( dist `  U
)  e.  ( * Met `  ( Base `  U ) )  /\  ( Base `  U )  C_  ( Base `  U
) )  ->  (
( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) )  e.  ( * Met `  ( Base `  U
) ) )
2421, 22, 23sylancl 643 . 2  |-  ( ph  ->  ( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) )  e.  ( * Met `  ( Base `  U ) ) )
25 eqid 2283 . . . 4  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
26 eqid 2283 . . . 4  |-  ( TopOpen `  U )  =  (
TopOpen `  U )
271, 2, 18, 4, 25, 26imastopn 17411 . . 3  |-  ( ph  ->  ( TopOpen `  U )  =  ( ( TopOpen `  R ) qTop  F )
)
2825, 7, 8xmstopn 17997 . . . . . 6  |-  ( R  e.  * MetSp  ->  ( TopOpen
`  R )  =  ( MetOpen `  ( ( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) )
294, 28syl 15 . . . . 5  |-  ( ph  ->  ( TopOpen `  R )  =  ( MetOpen `  (
( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) )
3012fveq2d 5529 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  =  ( MetOpen `  ( ( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) )
3129, 30eqtr4d 2318 . . . 4  |-  ( ph  ->  ( TopOpen `  R )  =  ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) ) )
3231oveq1d 5873 . . 3  |-  ( ph  ->  ( ( TopOpen `  R
) qTop  F )  =  ( ( MetOpen `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
) )
33 blbas 17976 . . . . . 6  |-  ( ( ( dist `  R
)  |`  ( V  X.  V ) )  e.  ( * Met `  V
)  ->  ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )  e.  TopBases )
3415, 33syl 15 . . . . 5  |-  ( ph  ->  ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) )  e.  TopBases )
35 unirnbl 17969 . . . . . . 7  |-  ( ( ( dist `  R
)  |`  ( V  X.  V ) )  e.  ( * Met `  V
)  ->  U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V ) ) )  =  V )
36 f1oeq2 5464 . . . . . . 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 ) )
3715, 35, 363syl 18 . . . . . 6  |-  ( ph  ->  ( F : U. ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) -1-1-onto-> B  <->  F : V
-1-1-onto-> B ) )
383, 37mpbird 223 . . . . 5  |-  ( ph  ->  F : U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) -1-1-onto-> B )
39 eqid 2283 . . . . . 6  |-  U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  = 
U. ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )
4039tgqtop 17403 . . . . 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 ) ) )
4134, 38, 40syl2anc 642 . . . 4  |-  ( ph  ->  ( ( topGen `  ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) ) qTop 
F )  =  (
topGen `  ( ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V ) ) ) qTop  F ) ) )
42 eqid 2283 . . . . . . 7  |-  ( MetOpen `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )  =  ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) )
4342mopnval 17984 . . . . . 6  |-  ( ( ( dist `  R
)  |`  ( V  X.  V ) )  e.  ( * Met `  V
)  ->  ( MetOpen `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )  =  ( topGen `  ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) ) )
4415, 43syl 15 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  =  ( topGen `  ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) ) )
4544oveq1d 5873 . . . 4  |-  ( ph  ->  ( ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
)  =  ( (
topGen `  ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) ) qTop  F ) )
46 eqid 2283 . . . . . . 7  |-  ( MetOpen `  ( dist `  U )
)  =  ( MetOpen `  ( dist `  U )
)
4746mopnval 17984 . . . . . 6  |-  ( (
dist `  U )  e.  ( * Met `  B
)  ->  ( MetOpen `  ( dist `  U )
)  =  ( topGen ` 
ran  ( ball `  ( dist `  U ) ) ) )
4816, 47syl 15 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( dist `  U ) )  =  ( topGen `  ran  ( ball `  ( dist `  U
) ) ) )
49 xmetf 17894 . . . . . . . 8  |-  ( (
dist `  U )  e.  ( * Met `  ( Base `  U ) )  ->  ( dist `  U
) : ( (
Base `  U )  X.  ( Base `  U
) ) --> RR* )
5021, 49syl 15 . . . . . . 7  |-  ( ph  ->  ( dist `  U
) : ( (
Base `  U )  X.  ( Base `  U
) ) --> RR* )
51 ffn 5389 . . . . . . 7  |-  ( (
dist `  U ) : ( ( Base `  U )  X.  ( Base `  U ) ) -->
RR*  ->  ( dist `  U
)  Fn  ( (
Base `  U )  X.  ( Base `  U
) ) )
52 fnresdm 5353 . . . . . . 7  |-  ( (
dist `  U )  Fn  ( ( Base `  U
)  X.  ( Base `  U ) )  -> 
( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) )  =  ( dist `  U ) )
5350, 51, 523syl 18 . . . . . 6  |-  ( ph  ->  ( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) )  =  ( dist `  U ) )
5453fveq2d 5529 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( ( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) ) )  =  ( MetOpen `  ( dist `  U )
) )
553ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  F : V -1-1-onto-> B
)
56 f1of1 5471 . . . . . . . . . . . . . . 15  |-  ( F : V -1-1-onto-> B  ->  F : V -1-1-> B )
5755, 56syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  F : V -1-1-> B )
58 cnvimass 5033 . . . . . . . . . . . . . . 15  |-  ( `' F " x ) 
C_  dom  F
59 f1odm 5476 . . . . . . . . . . . . . . . 16  |-  ( F : V -1-1-onto-> B  ->  dom  F  =  V )
6055, 59syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  dom  F  =  V )
6158, 60syl5sseq 3226 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( `' F " x )  C_  V
)
6215ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( ( dist `  R )  |`  ( V  X.  V ) )  e.  ( * Met `  V ) )
63 simprl 732 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  y  e.  V
)
64 simprr 733 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  r  e.  RR* )
65 blssm 17968 . . . . . . . . . . . . . . 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 )
6662, 63, 64, 65syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( y (
ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) r )  C_  V )
67 f1imaeq 5789 . . . . . . . . . . . . . 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 ) ) )
6857, 61, 66, 67syl12anc 1180 . . . . . . . . . . . . 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 ) ) )
6955, 17syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  F : V -onto-> B )
70 simplr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  x  C_  B
)
71 foimacnv 5490 . . . . . . . . . . . . . . 15  |-  ( ( F : V -onto-> B  /\  x  C_  B )  ->  ( F "
( `' F "
x ) )  =  x )
7269, 70, 71syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  ( F "
( `' F "
x ) )  =  x )
731ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  U  =  ( F  "s  R ) )
742ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  V  =  (
Base `  R )
)
754ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  C_  B )  /\  (
y  e.  V  /\  r  e.  RR* ) )  ->  R  e.  * MetSp )
7673, 74, 55, 75, 5, 6, 62, 63, 64imasf1obl 18034 . . . . . . . . . . . . . . 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 ) ) )
7776eqcomd 2288 . . . . . . . . . . . . . 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 ) )
7872, 77eqeq12d 2297 . . . . . . . . . . . . 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 ) ) )
7968, 78bitr3d 246 . . . . . . . . . . . 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 ) ) )
80792rexbidva 2584 . . . . . . . . . . 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 ) ) )
813adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  B
)  ->  F : V
-1-1-onto-> B )
82 f1ofn 5473 . . . . . . . . . . . 12  |-  ( F : V -1-1-onto-> B  ->  F  Fn  V )
83 oveq1 5865 . . . . . . . . . . . . . . 15  |-  ( z  =  ( F `  y )  ->  (
z ( ball `  ( dist `  U ) ) r )  =  ( ( F `  y
) ( ball `  ( dist `  U ) ) r ) )
8483eqeq2d 2294 . . . . . . . . . . . . . 14  |-  ( z  =  ( F `  y )  ->  (
x  =  ( z ( ball `  ( dist `  U ) ) r )  <->  x  =  ( ( F `  y ) ( ball `  ( dist `  U
) ) r ) ) )
8584rexbidv 2564 . . . . . . . . . . . . 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 ) ) )
8685rexrn 5667 . . . . . . . . . . . 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 ) ) )
8781, 82, 863syl 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 ) ) )
88 forn 5454 . . . . . . . . . . . . 13  |-  ( F : V -onto-> B  ->  ran  F  =  B )
8981, 17, 883syl 18 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  B
)  ->  ran  F  =  B )
9089rexeqdv 2743 . . . . . . . . . . 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 ) ) )
9180, 87, 903bitr2d 272 . . . . . . . . . 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 ) ) )
9215adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  B
)  ->  ( ( dist `  R )  |`  ( V  X.  V
) )  e.  ( * Met `  V
) )
93 blrn 17962 . . . . . . . . . . 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 ) ) )
9492, 93syl 15 . . . . . . . . . 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 ) ) )
9516adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  x  C_  B
)  ->  ( dist `  U )  e.  ( * Met `  B
) )
96 blrn 17962 . . . . . . . . . . 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 ) ) )
9795, 96syl 15 . . . . . . . . . 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 ) ) )
9891, 94, 973bitr4d 276 . . . . . . . . 9  |-  ( (
ph  /\  x  C_  B
)  ->  ( ( `' F " x )  e.  ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )  <-> 
x  e.  ran  ( ball `  ( dist `  U
) ) ) )
9998pm5.32da 622 . . . . . . . 8  |-  ( ph  ->  ( ( x  C_  B  /\  ( `' F " x )  e.  ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) )  <-> 
( x  C_  B  /\  x  e.  ran  ( ball `  ( dist `  U ) ) ) ) )
100 f1ofo 5479 . . . . . . . . . 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 )
10138, 100syl 15 . . . . . . . . 9  |-  ( ph  ->  F : U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) -onto-> B )
10239elqtop2 17392 . . . . . . . . 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 ) ) ) ) ) )
10334, 101, 102syl2anc 642 . . . . . . . 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
) ) ) ) ) )
104 blf 17961 . . . . . . . . . . . 12  |-  ( (
dist `  U )  e.  ( * Met `  B
)  ->  ( ball `  ( dist `  U
) ) : ( B  X.  RR* ) --> ~P B )
105 frn 5395 . . . . . . . . . . . 12  |-  ( (
ball `  ( dist `  U ) ) : ( B  X.  RR* )
--> ~P B  ->  ran  ( ball `  ( dist `  U ) )  C_  ~P B )
10616, 104, 1053syl 18 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( ball `  ( dist `  U ) ) 
C_  ~P B )
107106sseld 3179 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ran  ( ball `  ( dist `  U ) )  ->  x  e.  ~P B
) )
108 elpwi 3633 . . . . . . . . . 10  |-  ( x  e.  ~P B  ->  x  C_  B )
109107, 108syl6 29 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ran  ( ball `  ( dist `  U ) )  ->  x  C_  B ) )
110109pm4.71rd 616 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ran  ( ball `  ( dist `  U ) )  <->  ( x  C_  B  /\  x  e. 
ran  ( ball `  ( dist `  U ) ) ) ) )
11199, 103, 1103bitr4d 276 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( ran  ( ball `  (
( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
)  <->  x  e.  ran  ( ball `  ( dist `  U ) ) ) )
112111eqrdv 2281 . . . . . 6  |-  ( ph  ->  ( ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) qTop 
F )  =  ran  ( ball `  ( dist `  U ) ) )
113112fveq2d 5529 . . . . 5  |-  ( ph  ->  ( topGen `  ( ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
) )  =  (
topGen `  ran  ( ball `  ( dist `  U
) ) ) )
11448, 54, 1133eqtr4d 2325 . . . 4  |-  ( ph  ->  ( MetOpen `  ( ( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) ) )  =  ( topGen `  ( ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) qTop 
F ) ) )
11541, 45, 1143eqtr4d 2325 . . 3  |-  ( ph  ->  ( ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
)  =  ( MetOpen `  ( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) ) ) )
11627, 32, 1153eqtrd 2319 . 2  |-  ( ph  ->  ( TopOpen `  U )  =  ( MetOpen `  (
( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) ) ) )
117 eqid 2283 . . 3  |-  ( Base `  U )  =  (
Base `  U )
118 eqid 2283 . . 3  |-  ( (
dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) )  =  ( ( dist `  U )  |`  (
( Base `  U )  X.  ( Base `  U
) ) )
11926, 117, 118isxms2 17994 . 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 ) ) ) ) ) )
12024, 116, 119sylanbrc 645 1  |-  ( ph  ->  U  e.  * MetSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   ~Pcpw 3625   U.cuni 3827    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   RR*cxr 8866   Basecbs 13148   distcds 13217   TopOpenctopn 13326   topGenctg 13342   qTop cqtop 13406    "s cimas 13407   * Metcxmt 16369   ballcbl 16371   MetOpencmopn 16372   TopBasesctb 16635   * MetSpcxme 17882
This theorem is referenced by:  imasf1oms  18036  xpsxms  18080
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-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
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-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  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-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  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-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-xms 17885
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