MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imasf1obl Unicode version

Theorem imasf1obl 18401
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 5947 . . . . . . . . . 10  |-  ( ( F : V -1-1-onto-> B  /\  x  e.  B )  ->  ( F `  ( `' F `  x ) )  =  x )
31, 2sylan 458 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  ( F `  ( `' F `  x )
)  =  x )
43oveq2d 6029 . . . . . . . 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 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  U  =  ( F  "s  R
) )
7 imasf1obl.v . . . . . . . . . 10  |-  ( ph  ->  V  =  ( Base `  R ) )
87adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  V  =  ( Base `  R
) )
91adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  F : V -1-1-onto-> B )
10 imasf1obl.r . . . . . . . . . 10  |-  ( ph  ->  R  e.  Z )
1110adantr 452 . . . . . . . . 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 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  E  e.  ( * Met `  V
) )
16 imasf1obl.x . . . . . . . . . 10  |-  ( ph  ->  P  e.  V )
1716adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  P  e.  V )
18 f1ocnv 5620 . . . . . . . . . . . 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 5607 . . . . . . . . . . 11  |-  ( `' F : B -1-1-onto-> V  ->  `' F : B --> V )
2119, 20syl 16 . . . . . . . . . 10  |-  ( ph  ->  `' F : B --> V )
2221ffvelrnda 5802 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  ( `' F `  x )  e.  V )
236, 8, 9, 11, 12, 13, 15, 17, 22imasdsf1o 18305 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  (
( F `  P
) D ( F `
 ( `' F `  x ) ) )  =  ( P E ( `' F `  x ) ) )
244, 23eqtr3d 2414 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (
( F `  P
) D x )  =  ( P E ( `' F `  x ) ) )
2524breq1d 4156 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (
( ( F `  P ) D x )  <  S  <->  ( P E ( `' F `  x ) )  < 
S ) )
26 imasf1obl.s . . . . . . . 8  |-  ( ph  ->  S  e.  RR* )
2726adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  S  e.  RR* )
28 elbl2 18317 . . . . . . 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 1185 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (
( `' F `  x )  e.  ( P ( ball `  E
) S )  <->  ( P E ( `' F `  x ) )  < 
S ) )
3025, 29bitr4d 248 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  (
( ( F `  P ) D x )  <  S  <->  ( `' F `  x )  e.  ( P ( ball `  E ) S ) ) )
3130pm5.32da 623 . . . 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 18306 . . . . 5  |-  ( ph  ->  D  e.  ( * Met `  B ) )
33 f1of 5607 . . . . . . 7  |-  ( F : V -1-1-onto-> B  ->  F : V
--> B )
341, 33syl 16 . . . . . 6  |-  ( ph  ->  F : V --> B )
3534, 16ffvelrnd 5803 . . . . 5  |-  ( ph  ->  ( F `  P
)  e.  B )
36 elbl 18316 . . . . 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 1184 . . . 4  |-  ( ph  ->  ( x  e.  ( ( F `  P
) ( ball `  D
) S )  <->  ( x  e.  B  /\  (
( F `  P
) D x )  <  S ) ) )
38 f1ofn 5608 . . . . 5  |-  ( `' F : B -1-1-onto-> V  ->  `' F  Fn  B
)
39 elpreima 5782 . . . . 5  |-  ( `' F  Fn  B  -> 
( x  e.  ( `' `' F " ( P ( ball `  E
) S ) )  <-> 
( x  e.  B  /\  ( `' F `  x )  e.  ( P ( ball `  E
) S ) ) ) )
4019, 38, 393syl 19 . . . 4  |-  ( ph  ->  ( x  e.  ( `' `' F " ( P ( ball `  E
) S ) )  <-> 
( x  e.  B  /\  ( `' F `  x )  e.  ( P ( ball `  E
) S ) ) ) )
4131, 37, 403bitr4d 277 . . 3  |-  ( ph  ->  ( x  e.  ( ( F `  P
) ( ball `  D
) S )  <->  x  e.  ( `' `' F " ( P ( ball `  E
) S ) ) ) )
4241eqrdv 2378 . 2  |-  ( ph  ->  ( ( F `  P ) ( ball `  D ) S )  =  ( `' `' F " ( P (
ball `  E ) S ) ) )
43 imacnvcnv 5267 . 2  |-  ( `' `' F " ( P ( ball `  E
) S ) )  =  ( F "
( P ( ball `  E ) S ) )
4442, 43syl6eq 2428 1  |-  ( ph  ->  ( ( F `  P ) ( ball `  D ) S )  =  ( F "
( P ( ball `  E ) S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4146    X. cxp 4809   `'ccnv 4810    |` cres 4813   "cima 4814    Fn wfn 5382   -->wf 5383   -1-1-onto->wf1o 5386   ` cfv 5387  (class class class)co 6013   RR*cxr 9045    < clt 9046   Basecbs 13389   distcds 13458    "s cimas 13650   * Metcxmt 16605   ballcbl 16607
This theorem is referenced by:  imasf1oxms  18402
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-rp 10538  df-xneg 10635  df-xadd 10636  df-xmul 10637  df-fz 10969  df-fzo 11059  df-seq 11244  df-hash 11539  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-sca 13465  df-vsca 13466  df-tset 13468  df-ple 13469  df-ds 13471  df-xrs 13646  df-0g 13647  df-gsum 13648  df-imas 13654  df-mre 13731  df-mrc 13732  df-acs 13734  df-mnd 14610  df-submnd 14659  df-mulg 14735  df-cntz 15036  df-cmn 15334  df-xmet 16612  df-bl 16614
  Copyright terms: Public domain W3C validator