Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rfcnpre1 Structured version   Unicode version

Theorem rfcnpre1 27680
Description: If F is a continuous function with respect to the standard topology, then the preimage A of the values greater than a given extended real B is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
rfcnpre1.1  |-  F/_ x B
rfcnpre1.2  |-  F/_ x F
rfcnpre1.3  |-  F/ x ph
rfcnpre1.4  |-  K  =  ( topGen `  ran  (,) )
rfcnpre1.5  |-  X  = 
U. J
rfcnpre1.6  |-  A  =  { x  e.  X  |  B  <  ( F `
 x ) }
rfcnpre1.7  |-  ( ph  ->  B  e.  RR* )
rfcnpre1.8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
Assertion
Ref Expression
rfcnpre1  |-  ( ph  ->  A  e.  J )

Proof of Theorem rfcnpre1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 rfcnpre1.3 . . . . 5  |-  F/ x ph
2 rfcnpre1.8 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
3 cntop1 17309 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
42, 3syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  J  e.  Top )
5 rfcnpre1.5 . . . . . . . . . . . . . 14  |-  X  = 
U. J
64, 5jctir 526 . . . . . . . . . . . . 13  |-  ( ph  ->  ( J  e.  Top  /\  X  =  U. J
) )
7 istopon 16995 . . . . . . . . . . . . 13  |-  ( J  e.  (TopOn `  X
)  <->  ( J  e. 
Top  /\  X  =  U. J ) )
86, 7sylibr 205 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  (TopOn `  X ) )
9 rfcnpre1.4 . . . . . . . . . . . . 13  |-  K  =  ( topGen `  ran  (,) )
10 retopon 18802 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
119, 10eqeltri 2508 . . . . . . . . . . . 12  |-  K  e.  (TopOn `  RR )
12 iscn 17304 . . . . . . . . . . . 12  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> RR  /\  A. y  e.  K  ( `' F " y )  e.  J ) ) )
138, 11, 12sylancl 645 . . . . . . . . . . 11  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
( F : X --> RR  /\  A. y  e.  K  ( `' F " y )  e.  J
) ) )
142, 13mpbid 203 . . . . . . . . . 10  |-  ( ph  ->  ( F : X --> RR  /\  A. y  e.  K  ( `' F " y )  e.  J
) )
1514simpld 447 . . . . . . . . 9  |-  ( ph  ->  F : X --> RR )
1615fnvinran 27675 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  RR )
17 rfcnpre1.7 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR* )
18 elioopnf 11003 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  ( ( F `  x )  e.  ( B (,)  +oo )  <->  ( ( F `
 x )  e.  RR  /\  B  < 
( F `  x
) ) ) )
1917, 18syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  x )  e.  ( B (,)  +oo )  <->  ( ( F `  x
)  e.  RR  /\  B  <  ( F `  x ) ) ) )
2019baibd 877 . . . . . . . 8  |-  ( (
ph  /\  ( F `  x )  e.  RR )  ->  ( ( F `
 x )  e.  ( B (,)  +oo ) 
<->  B  <  ( F `
 x ) ) )
2116, 20syldan 458 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( F `  x
)  e.  ( B (,)  +oo )  <->  B  <  ( F `  x ) ) )
2221pm5.32da 624 . . . . . 6  |-  ( ph  ->  ( ( x  e.  X  /\  ( F `
 x )  e.  ( B (,)  +oo ) )  <->  ( x  e.  X  /\  B  < 
( F `  x
) ) ) )
23 ffn 5594 . . . . . . 7  |-  ( F : X --> RR  ->  F  Fn  X )
24 elpreima 5853 . . . . . . 7  |-  ( F  Fn  X  ->  (
x  e.  ( `' F " ( B (,)  +oo ) )  <->  ( x  e.  X  /\  ( F `  x )  e.  ( B (,)  +oo ) ) ) )
2515, 23, 243syl 19 . . . . . 6  |-  ( ph  ->  ( x  e.  ( `' F " ( B (,)  +oo ) )  <->  ( x  e.  X  /\  ( F `  x )  e.  ( B (,)  +oo ) ) ) )
26 rabid 2886 . . . . . . 7  |-  ( x  e.  { x  e.  X  |  B  < 
( F `  x
) }  <->  ( x  e.  X  /\  B  < 
( F `  x
) ) )
2726a1i 11 . . . . . 6  |-  ( ph  ->  ( x  e.  {
x  e.  X  |  B  <  ( F `  x ) }  <->  ( x  e.  X  /\  B  < 
( F `  x
) ) ) )
2822, 25, 273bitr4d 278 . . . . 5  |-  ( ph  ->  ( x  e.  ( `' F " ( B (,)  +oo ) )  <->  x  e.  { x  e.  X  |  B  <  ( F `  x ) } ) )
291, 28alrimi 1782 . . . 4  |-  ( ph  ->  A. x ( x  e.  ( `' F " ( B (,)  +oo ) )  <->  x  e.  { x  e.  X  |  B  <  ( F `  x ) } ) )
30 rfcnpre1.2 . . . . . . 7  |-  F/_ x F
3130nfcnv 5054 . . . . . 6  |-  F/_ x `' F
32 rfcnpre1.1 . . . . . . 7  |-  F/_ x B
33 nfcv 2574 . . . . . . 7  |-  F/_ x (,)
34 nfcv 2574 . . . . . . 7  |-  F/_ x  +oo
3532, 33, 34nfov 6107 . . . . . 6  |-  F/_ x
( B (,)  +oo )
3631, 35nfima 5214 . . . . 5  |-  F/_ x
( `' F "
( B (,)  +oo ) )
37 nfrab1 2890 . . . . 5  |-  F/_ x { x  e.  X  |  B  <  ( F `
 x ) }
3836, 37cleqf 2598 . . . 4  |-  ( ( `' F " ( B (,)  +oo ) )  =  { x  e.  X  |  B  <  ( F `
 x ) }  <->  A. x ( x  e.  ( `' F "
( B (,)  +oo ) )  <->  x  e.  { x  e.  X  |  B  <  ( F `  x ) } ) )
3929, 38sylibr 205 . . 3  |-  ( ph  ->  ( `' F "
( B (,)  +oo ) )  =  {
x  e.  X  |  B  <  ( F `  x ) } )
40 rfcnpre1.6 . . 3  |-  A  =  { x  e.  X  |  B  <  ( F `
 x ) }
4139, 40syl6eqr 2488 . 2  |-  ( ph  ->  ( `' F "
( B (,)  +oo ) )  =  A )
42 iooretop 18805 . . . 4  |-  ( B (,)  +oo )  e.  (
topGen `  ran  (,) )
4342, 9eleqtrri 2511 . . 3  |-  ( B (,)  +oo )  e.  K
44 cnima 17334 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( B (,)  +oo )  e.  K )  ->  ( `' F " ( B (,)  +oo ) )  e.  J )
452, 43, 44sylancl 645 . 2  |-  ( ph  ->  ( `' F "
( B (,)  +oo ) )  e.  J
)
4641, 45eqeltrrd 2513 1  |-  ( ph  ->  A  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550   F/wnf 1554    = wceq 1653    e. wcel 1726   F/_wnfc 2561   A.wral 2707   {crab 2711   U.cuni 4017   class class class wbr 4215   `'ccnv 4880   ran crn 4882   "cima 4884    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084   RRcr 8994    +oocpnf 9122   RR*cxr 9124    < clt 9125   (,)cioo 10921   topGenctg 13670   Topctop 16963  TopOnctopon 16964    Cn ccn 17293
This theorem is referenced by:  stoweidlem46  27785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-q 10580  df-ioo 10925  df-topgen 13672  df-top 16968  df-bases 16970  df-topon 16971  df-cn 17296
  Copyright terms: Public domain W3C validator