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Theorem imasf1obl 21116
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 6184 . . . . . . . . . 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 6312 . . . . . . . 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 21002 . . . . . . . 8  |-  ( (
ph  /\  x  e.  B )  ->  (
( F `  P
) D ( F `
 ( `' F `  x ) ) )  =  ( P E ( `' F `  x ) ) )
244, 23eqtr3d 2500 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  (
( F `  P
) D x )  =  ( P E ( `' F `  x ) ) )
2524breq1d 4466 . . . . . 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 21018 . . . . . . 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 21003 . . . . 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 21016 . . . . 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 2454 . 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 2514 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 1395    e. wcel 1819   class class class wbr 4456    X. cxp 5006   `'ccnv 5007    |` cres 5010   "cima 5011    Fn wfn 5589   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   RR*cxr 9644    < clt 9645   Basecbs 14643   distcds 14720    "s cimas 14920   *Metcxmt 18529   ballcbl 18531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-fz 11698  df-fzo 11821  df-seq 12110  df-hash 12408  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-0g 14858  df-gsum 14859  df-xrs 14918  df-imas 14924  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-mulg 16186  df-cntz 16481  df-cmn 16926  df-psmet 18537  df-xmet 18538  df-bl 18540
This theorem is referenced by:  imasf1oxms  21117
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