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Theorem imasf1obl 20068
Description: The image of a metric space ball. (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 )
imasf1obl.r  |-  ( ph  ->  R  e.  Z )
imasf1obl.e  |-  E  =  ( ( dist `  R
)  |`  ( V  X.  V ) )
imasf1obl.d  |-  D  =  ( dist `  U
)
imasf1obl.m  |-  ( ph  ->  E  e.  ( *Met `  V ) )
imasf1obl.x  |-  ( ph  ->  P  e.  V )
imasf1obl.s  |-  ( ph  ->  S  e.  RR* )
Assertion
Ref Expression
imasf1obl  |-  ( ph  ->  ( ( F `  P ) ( ball `  D ) S )  =  ( F "
( P ( ball `  E ) S ) ) )

Proof of Theorem imasf1obl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 imasf1obl.f . . . . . . . . . 10  |-  ( ph  ->  F : V -1-1-onto-> B )
2 f1ocnvfv2 5989 . . . . . . . . . 10  |-  ( ( F : V -1-1-onto-> B  /\  x  e.  B )  ->  ( F `  ( `' F `  x ) )  =  x )
31, 2sylan 471 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  ( F `  ( `' F `  x )
)  =  x )
43oveq2d 6112 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  (
( F `  P
) D ( F `
 ( `' F `  x ) ) )  =  ( ( F `
 P ) D x ) )
5 imasf1obl.u . . . . . . . . . 10  |-  ( ph  ->  U  =  ( F 
"s  R ) )
65adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  U  =  ( F  "s  R
) )
7 imasf1obl.v . . . . . . . . . 10  |-  ( ph  ->  V  =  ( Base `  R ) )
87adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  V  =  ( Base `  R
) )
91adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  F : V -1-1-onto-> B )
10 imasf1obl.r . . . . . . . . . 10  |-  ( ph  ->  R  e.  Z )
1110adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  R  e.  Z )
12 imasf1obl.e . . . . . . . . 9  |-  E  =  ( ( dist `  R
)  |`  ( V  X.  V ) )
13 imasf1obl.d . . . . . . . . 9  |-  D  =  ( dist `  U
)
14 imasf1obl.m . . . . . . . . . 10  |-  ( ph  ->  E  e.  ( *Met `  V ) )
1514adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  E  e.  ( *Met `  V ) )
16 imasf1obl.x . . . . . . . . . 10  |-  ( ph  ->  P  e.  V )
1716adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  P  e.  V )
18 f1ocnv 5658 . . . . . . . . . . . 12  |-  ( F : V -1-1-onto-> B  ->  `' F : B -1-1-onto-> V )
191, 18syl 16 . . . . . . . . . . 11  |-  ( ph  ->  `' F : B -1-1-onto-> V )
20 f1of 5646 . . . . . . . . . . 11  |-  ( `' F : B -1-1-onto-> V  ->  `' F : B --> V )
2119, 20syl 16 . . . . . . . . . 10  |-  ( ph  ->  `' F : B --> V )
2221ffvelrnda 5848 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  ( `' F `  x )  e.  V )
236, 8, 9, 11, 12, 13, 15, 17, 22imasdsf1o 19954 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  (
( F `  P
) D ( F `
 ( `' F `  x ) ) )  =  ( P E ( `' F `  x ) ) )
244, 23eqtr3d 2477 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (
( F `  P
) D x )  =  ( P E ( `' F `  x ) ) )
2524breq1d 4307 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (
( ( F `  P ) D x )  <  S  <->  ( P E ( `' F `  x ) )  < 
S ) )
26 imasf1obl.s . . . . . . . 8  |-  ( ph  ->  S  e.  RR* )
2726adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  S  e.  RR* )
28 elbl2 19970 . . . . . . 7  |-  ( ( ( E  e.  ( *Met `  V
)  /\  S  e.  RR* )  /\  ( P  e.  V  /\  ( `' F `  x )  e.  V ) )  ->  ( ( `' F `  x )  e.  ( P (
ball `  E ) S )  <->  ( P E ( `' F `  x ) )  < 
S ) )
2915, 27, 17, 22, 28syl22anc 1219 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (
( `' F `  x )  e.  ( P ( ball `  E
) S )  <->  ( P E ( `' F `  x ) )  < 
S ) )
3025, 29bitr4d 256 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  (
( ( F `  P ) D x )  <  S  <->  ( `' F `  x )  e.  ( P ( ball `  E ) S ) ) )
3130pm5.32da 641 . . . 4  |-  ( ph  ->  ( ( x  e.  B  /\  ( ( F `  P ) D x )  < 
S )  <->  ( x  e.  B  /\  ( `' F `  x )  e.  ( P (
ball `  E ) S ) ) ) )
325, 7, 1, 10, 12, 13, 14imasf1oxmet 19955 . . . . 5  |-  ( ph  ->  D  e.  ( *Met `  B ) )
33 f1of 5646 . . . . . . 7  |-  ( F : V -1-1-onto-> B  ->  F : V
--> B )
341, 33syl 16 . . . . . 6  |-  ( ph  ->  F : V --> B )
3534, 16ffvelrnd 5849 . . . . 5  |-  ( ph  ->  ( F `  P
)  e.  B )
36 elbl 19968 . . . . 5  |-  ( ( D  e.  ( *Met `  B )  /\  ( F `  P )  e.  B  /\  S  e.  RR* )  ->  ( x  e.  ( ( F `  P
) ( ball `  D
) S )  <->  ( x  e.  B  /\  (
( F `  P
) D x )  <  S ) ) )
3732, 35, 26, 36syl3anc 1218 . . . 4  |-  ( ph  ->  ( x  e.  ( ( F `  P
) ( ball `  D
) S )  <->  ( x  e.  B  /\  (
( F `  P
) D x )  <  S ) ) )
38 f1ofn 5647 . . . . 5  |-  ( `' F : B -1-1-onto-> V  ->  `' F  Fn  B
)
39 elpreima 5828 . . . . 5  |-  ( `' F  Fn  B  -> 
( x  e.  ( `' `' F " ( P ( ball `  E
) S ) )  <-> 
( x  e.  B  /\  ( `' F `  x )  e.  ( P ( ball `  E
) S ) ) ) )
4019, 38, 393syl 20 . . . 4  |-  ( ph  ->  ( x  e.  ( `' `' F " ( P ( ball `  E
) S ) )  <-> 
( x  e.  B  /\  ( `' F `  x )  e.  ( P ( ball `  E
) S ) ) ) )
4131, 37, 403bitr4d 285 . . 3  |-  ( ph  ->  ( x  e.  ( ( F `  P
) ( ball `  D
) S )  <->  x  e.  ( `' `' F " ( P ( ball `  E
) S ) ) ) )
4241eqrdv 2441 . 2  |-  ( ph  ->  ( ( F `  P ) ( ball `  D ) S )  =  ( `' `' F " ( P (
ball `  E ) S ) ) )
43 imacnvcnv 5308 . 2  |-  ( `' `' F " ( P ( ball `  E
) S ) )  =  ( F "
( P ( ball `  E ) S ) )
4442, 43syl6eq 2491 1  |-  ( ph  ->  ( ( F `  P ) ( ball `  D ) S )  =  ( F "
( P ( ball `  E ) S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4297    X. cxp 4843   `'ccnv 4844    |` cres 4847   "cima 4848    Fn wfn 5418   -->wf 5419   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096   RR*cxr 9422    < clt 9423   Basecbs 14179   distcds 14252    "s cimas 14447   *Metcxmt 17806   ballcbl 17808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-0g 14385  df-gsum 14386  df-xrs 14445  df-imas 14451  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284  df-psmet 17814  df-xmet 17815  df-bl 17817
This theorem is referenced by:  imasf1oxms  20069
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