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Theorem imasf1obl 20859
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 6182 . . . . . . . . . 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 6311 . . . . . . . 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 5834 . . . . . . . . . . . 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 5822 . . . . . . . . . . 11  |-  ( `' F : B -1-1-onto-> V  ->  `' F : B --> V )
2119, 20syl 16 . . . . . . . . . 10  |-  ( ph  ->  `' F : B --> V )
2221ffvelrnda 6032 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  ( `' F `  x )  e.  V )
236, 8, 9, 11, 12, 13, 15, 17, 22imasdsf1o 20745 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  (
( F `  P
) D ( F `
 ( `' F `  x ) ) )  =  ( P E ( `' F `  x ) ) )
244, 23eqtr3d 2510 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (
( F `  P
) D x )  =  ( P E ( `' F `  x ) ) )
2524breq1d 4463 . . . . . 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 20761 . . . . . . 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 1229 . . . . . 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 20746 . . . . 5  |-  ( ph  ->  D  e.  ( *Met `  B ) )
33 f1of 5822 . . . . . . 7  |-  ( F : V -1-1-onto-> B  ->  F : V
--> B )
341, 33syl 16 . . . . . 6  |-  ( ph  ->  F : V --> B )
3534, 16ffvelrnd 6033 . . . . 5  |-  ( ph  ->  ( F `  P
)  e.  B )
36 elbl 20759 . . . . 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 1228 . . . 4  |-  ( ph  ->  ( x  e.  ( ( F `  P
) ( ball `  D
) S )  <->  ( x  e.  B  /\  (
( F `  P
) D x )  <  S ) ) )
38 f1ofn 5823 . . . . 5  |-  ( `' F : B -1-1-onto-> V  ->  `' F  Fn  B
)
39 elpreima 6008 . . . . 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 2464 . 2  |-  ( ph  ->  ( ( F `  P ) ( ball `  D ) S )  =  ( `' `' F " ( P (
ball `  E ) S ) ) )
43 imacnvcnv 5478 . 2  |-  ( `' `' F " ( P ( ball `  E
) S ) )  =  ( F "
( P ( ball `  E ) S ) )
4442, 43syl6eq 2524 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 1379    e. wcel 1767   class class class wbr 4453    X. cxp 5003   `'ccnv 5004    |` cres 5007   "cima 5008    Fn wfn 5589   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   RR*cxr 9639    < clt 9640   Basecbs 14507   distcds 14581    "s cimas 14776   *Metcxmt 18273   ballcbl 18275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-0g 14714  df-gsum 14715  df-xrs 14774  df-imas 14780  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-bl 18284
This theorem is referenced by:  imasf1oxms  20860
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