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Theorem imasf1oxms 18051
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 2296 . . . . 5  |-  ( (
dist `  R )  |`  ( V  X.  V
) )  =  ( ( dist `  R
)  |`  ( V  X.  V ) )
6 eqid 2296 . . . . 5  |-  ( dist `  U )  =  (
dist `  U )
7 eqid 2296 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
8 eqid 2296 . . . . . . . 8  |-  ( (
dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) )  =  ( ( dist `  R )  |`  (
( Base `  R )  X.  ( Base `  R
) ) )
97, 8xmsxmet 18018 . . . . . . 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 4730 . . . . . . . 8  |-  ( ph  ->  ( V  X.  V
)  =  ( (
Base `  R )  X.  ( Base `  R
) ) )
1211reseq2d 4971 . . . . . . 7  |-  ( ph  ->  ( ( dist `  R
)  |`  ( V  X.  V ) )  =  ( ( dist `  R
)  |`  ( ( Base `  R )  X.  ( Base `  R ) ) ) )
132fveq2d 5545 . . . . . . 7  |-  ( ph  ->  ( * Met `  V
)  =  ( * Met `  ( Base `  R ) ) )
1412, 13eleq12d 2364 . . . . . 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 17955 . . . 4  |-  ( ph  ->  ( dist `  U
)  e.  ( * Met `  B ) )
17 f1ofo 5495 . . . . . . 7  |-  ( F : V -1-1-onto-> B  ->  F : V -onto-> B )
183, 17syl 15 . . . . . 6  |-  ( ph  ->  F : V -onto-> B
)
191, 2, 18, 4imasbas 13431 . . . . 5  |-  ( ph  ->  B  =  ( Base `  U ) )
2019fveq2d 5545 . . . 4  |-  ( ph  ->  ( * Met `  B
)  =  ( * Met `  ( Base `  U ) ) )
2116, 20eleqtrd 2372 . . 3  |-  ( ph  ->  ( dist `  U
)  e.  ( * Met `  ( Base `  U ) ) )
22 ssid 3210 . . 3  |-  ( Base `  U )  C_  ( Base `  U )
23 xmetres2 17941 . . 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 2296 . . . 4  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
26 eqid 2296 . . . 4  |-  ( TopOpen `  U )  =  (
TopOpen `  U )
271, 2, 18, 4, 25, 26imastopn 17427 . . 3  |-  ( ph  ->  ( TopOpen `  U )  =  ( ( TopOpen `  R ) qTop  F )
)
2825, 7, 8xmstopn 18013 . . . . . 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 5545 . . . . 5  |-  ( ph  ->  ( MetOpen `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  =  ( MetOpen `  ( ( dist `  R )  |`  ( ( Base `  R
)  X.  ( Base `  R ) ) ) ) )
3129, 30eqtr4d 2331 . . . 4  |-  ( ph  ->  ( TopOpen `  R )  =  ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) ) )
3231oveq1d 5889 . . 3  |-  ( ph  ->  ( ( TopOpen `  R
) qTop  F )  =  ( ( MetOpen `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
) )
33 blbas 17992 . . . . . 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 17985 . . . . . . 7  |-  ( ( ( dist `  R
)  |`  ( V  X.  V ) )  e.  ( * Met `  V
)  ->  U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V ) ) )  =  V )
36 f1oeq2 5480 . . . . . . 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 2296 . . . . . 6  |-  U. ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) )  = 
U. ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )
4039tgqtop 17419 . . . . 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 2296 . . . . . . 7  |-  ( MetOpen `  ( ( dist `  R
)  |`  ( V  X.  V ) ) )  =  ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) )
4342mopnval 18000 . . . . . 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 5889 . . . 4  |-  ( ph  ->  ( ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
)  =  ( (
topGen `  ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) ) qTop  F ) )
46 eqid 2296 . . . . . . 7  |-  ( MetOpen `  ( dist `  U )
)  =  ( MetOpen `  ( dist `  U )
)
4746mopnval 18000 . . . . . 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 17910 . . . . . . . 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 5405 . . . . . . 7  |-  ( (
dist `  U ) : ( ( Base `  U )  X.  ( Base `  U ) ) -->
RR*  ->  ( dist `  U
)  Fn  ( (
Base `  U )  X.  ( Base `  U
) ) )
52 fnresdm 5369 . . . . . . 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 5545 . . . . 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 5487 . . . . . . . . . . . . . . 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 5049 . . . . . . . . . . . . . . 15  |-  ( `' F " x ) 
C_  dom  F
59 f1odm 5492 . . . . . . . . . . . . . . . 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 3239 . . . . . . . . . . . . . 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 17984 . . . . . . . . . . . . . . 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 5805 . . . . . . . . . . . . . 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 5506 . . . . . . . . . . . . . . 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 18050 . . . . . . . . . . . . . . 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 2301 . . . . . . . . . . . . . 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 2310 . . . . . . . . . . . . 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 2597 . . . . . . . . . . 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 5489 . . . . . . . . . . . 12  |-  ( F : V -1-1-onto-> B  ->  F  Fn  V )
83 oveq1 5881 . . . . . . . . . . . . . . 15  |-  ( z  =  ( F `  y )  ->  (
z ( ball `  ( dist `  U ) ) r )  =  ( ( F `  y
) ( ball `  ( dist `  U ) ) r ) )
8483eqeq2d 2307 . . . . . . . . . . . . . 14  |-  ( z  =  ( F `  y )  ->  (
x  =  ( z ( ball `  ( dist `  U ) ) r )  <->  x  =  ( ( F `  y ) ( ball `  ( dist `  U
) ) r ) ) )
8584rexbidv 2577 . . . . . . . . . . . . 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 5683 . . . . . . . . . . . 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 5470 . . . . . . . . . . . . 13  |-  ( F : V -onto-> B  ->  ran  F  =  B )
8981, 17, 883syl 18 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  C_  B
)  ->  ran  F  =  B )
9089rexeqdv 2756 . . . . . . . . . . 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 17978 . . . . . . . . . . 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 17978 . . . . . . . . . . 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 5495 . . . . . . . . . 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 17408 . . . . . . . . 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 17977 . . . . . . . . . . . 12  |-  ( (
dist `  U )  e.  ( * Met `  B
)  ->  ( ball `  ( dist `  U
) ) : ( B  X.  RR* ) --> ~P B )
105 frn 5411 . . . . . . . . . . . 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 3192 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ran  ( ball `  ( dist `  U ) )  ->  x  e.  ~P B
) )
108 elpwi 3646 . . . . . . . . . 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 2294 . . . . . 6  |-  ( ph  ->  ( ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) qTop 
F )  =  ran  ( ball `  ( dist `  U ) ) )
113112fveq2d 5545 . . . . 5  |-  ( ph  ->  ( topGen `  ( ran  ( ball `  ( ( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
) )  =  (
topGen `  ran  ( ball `  ( dist `  U
) ) ) )
11448, 54, 1133eqtr4d 2338 . . . 4  |-  ( ph  ->  ( MetOpen `  ( ( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) ) )  =  ( topGen `  ( ran  ( ball `  ( ( dist `  R
)  |`  ( V  X.  V ) ) ) qTop 
F ) ) )
11541, 45, 1143eqtr4d 2338 . . 3  |-  ( ph  ->  ( ( MetOpen `  (
( dist `  R )  |`  ( V  X.  V
) ) ) qTop  F
)  =  ( MetOpen `  ( ( dist `  U
)  |`  ( ( Base `  U )  X.  ( Base `  U ) ) ) ) )
11627, 32, 1153eqtrd 2332 . 2  |-  ( ph  ->  ( TopOpen `  U )  =  ( MetOpen `  (
( dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) ) ) )
117 eqid 2296 . . 3  |-  ( Base `  U )  =  (
Base `  U )
118 eqid 2296 . . 3  |-  ( (
dist `  U )  |`  ( ( Base `  U
)  X.  ( Base `  U ) ) )  =  ( ( dist `  U )  |`  (
( Base `  U )  X.  ( Base `  U
) ) )
11926, 117, 118isxms2 18010 . 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 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   ~Pcpw 3638   U.cuni 3843    X. cxp 4703   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708    Fn wfn 5266   -->wf 5267   -1-1->wf1 5268   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   RR*cxr 8882   Basecbs 13164   distcds 13233   TopOpenctopn 13342   topGenctg 13358   qTop cqtop 13422    "s cimas 13423   * Metcxmt 16385   ballcbl 16387   MetOpencmopn 16388   TopBasesctb 16651   * MetSpcxme 17898
This theorem is referenced by:  imasf1oms  18052  xpsxms  18096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-xms 17901
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