Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rfcnpre4 Structured version   Unicode version

Theorem rfcnpre4 29756
Description: If F is a continuous function with respect to the standard topology, then the preimage A of the values smaller or equal than a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
rfcnpre4.1  |-  F/_ t F
rfcnpre4.2  |-  K  =  ( topGen `  ran  (,) )
rfcnpre4.3  |-  T  = 
U. J
rfcnpre4.4  |-  A  =  { t  e.  T  |  ( F `  t )  <_  B }
rfcnpre4.5  |-  ( ph  ->  B  e.  RR )
rfcnpre4.6  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
Assertion
Ref Expression
rfcnpre4  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
Distinct variable groups:    t, B    t, T
Allowed substitution hints:    ph( t)    A( t)    F( t)    J( t)    K( t)

Proof of Theorem rfcnpre4
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 rfcnpre4.2 . . . . . . . 8  |-  K  =  ( topGen `  ran  (,) )
2 rfcnpre4.3 . . . . . . . 8  |-  T  = 
U. J
3 eqid 2443 . . . . . . . 8  |-  ( J  Cn  K )  =  ( J  Cn  K
)
4 rfcnpre4.6 . . . . . . . 8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
51, 2, 3, 4fcnre 29747 . . . . . . 7  |-  ( ph  ->  F : T --> RR )
6 ffn 5559 . . . . . . 7  |-  ( F : T --> RR  ->  F  Fn  T )
7 elpreima 5823 . . . . . . 7  |-  ( F  Fn  T  ->  (
s  e.  ( `' F " ( -oo (,] B ) )  <->  ( s  e.  T  /\  ( F `  s )  e.  ( -oo (,] B
) ) ) )
85, 6, 73syl 20 . . . . . 6  |-  ( ph  ->  ( s  e.  ( `' F " ( -oo (,] B ) )  <->  ( s  e.  T  /\  ( F `  s )  e.  ( -oo (,] B
) ) ) )
9 mnfxr 11094 . . . . . . . . 9  |- -oo  e.  RR*
10 rfcnpre4.5 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR )
1110rexrd 9433 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR* )
1211adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  T )  ->  B  e.  RR* )
13 elioc1 11342 . . . . . . . . 9  |-  ( ( -oo  e.  RR*  /\  B  e.  RR* )  ->  (
( F `  s
)  e.  ( -oo (,] B )  <->  ( ( F `  s )  e.  RR*  /\ -oo  <  ( F `  s )  /\  ( F `  s )  <_  B
) ) )
149, 12, 13sylancr 663 . . . . . . . 8  |-  ( (
ph  /\  s  e.  T )  ->  (
( F `  s
)  e.  ( -oo (,] B )  <->  ( ( F `  s )  e.  RR*  /\ -oo  <  ( F `  s )  /\  ( F `  s )  <_  B
) ) )
15 simpr3 996 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  T )  /\  (
( F `  s
)  e.  RR*  /\ -oo  <  ( F `  s
)  /\  ( F `  s )  <_  B
) )  ->  ( F `  s )  <_  B )
165fnvinran 29736 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  T )  ->  ( F `  s )  e.  RR )
1716rexrd 9433 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  T )  ->  ( F `  s )  e.  RR* )
1817adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  ( F `  s )  e.  RR* )
1916adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  ( F `  s )  e.  RR )
20 mnflt 11104 . . . . . . . . . . 11  |-  ( ( F `  s )  e.  RR  -> -oo  <  ( F `  s ) )
2119, 20syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  -> -oo  <  ( F `  s ) )
22 simpr 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  ( F `  s )  <_  B )
2318, 21, 223jca 1168 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  (
( F `  s
)  e.  RR*  /\ -oo  <  ( F `  s
)  /\  ( F `  s )  <_  B
) )
2415, 23impbida 828 . . . . . . . 8  |-  ( (
ph  /\  s  e.  T )  ->  (
( ( F `  s )  e.  RR*  /\ -oo  <  ( F `  s )  /\  ( F `  s )  <_  B )  <->  ( F `  s )  <_  B
) )
2514, 24bitrd 253 . . . . . . 7  |-  ( (
ph  /\  s  e.  T )  ->  (
( F `  s
)  e.  ( -oo (,] B )  <->  ( F `  s )  <_  B
) )
2625pm5.32da 641 . . . . . 6  |-  ( ph  ->  ( ( s  e.  T  /\  ( F `
 s )  e.  ( -oo (,] B
) )  <->  ( s  e.  T  /\  ( F `  s )  <_  B ) ) )
278, 26bitrd 253 . . . . 5  |-  ( ph  ->  ( s  e.  ( `' F " ( -oo (,] B ) )  <->  ( s  e.  T  /\  ( F `  s )  <_  B ) ) )
28 nfcv 2579 . . . . . 6  |-  F/_ t
s
29 nfcv 2579 . . . . . 6  |-  F/_ t T
30 rfcnpre4.1 . . . . . . . 8  |-  F/_ t F
3130, 28nffv 5698 . . . . . . 7  |-  F/_ t
( F `  s
)
32 nfcv 2579 . . . . . . 7  |-  F/_ t  <_
33 nfcv 2579 . . . . . . 7  |-  F/_ t B
3431, 32, 33nfbr 4336 . . . . . 6  |-  F/ t ( F `  s
)  <_  B
35 fveq2 5691 . . . . . . 7  |-  ( t  =  s  ->  ( F `  t )  =  ( F `  s ) )
3635breq1d 4302 . . . . . 6  |-  ( t  =  s  ->  (
( F `  t
)  <_  B  <->  ( F `  s )  <_  B
) )
3728, 29, 34, 36elrabf 3115 . . . . 5  |-  ( s  e.  { t  e.  T  |  ( F `
 t )  <_  B }  <->  ( s  e.  T  /\  ( F `
 s )  <_  B ) )
3827, 37syl6bbr 263 . . . 4  |-  ( ph  ->  ( s  e.  ( `' F " ( -oo (,] B ) )  <->  s  e.  { t  e.  T  | 
( F `  t
)  <_  B }
) )
3938eqrdv 2441 . . 3  |-  ( ph  ->  ( `' F "
( -oo (,] B ) )  =  { t  e.  T  |  ( F `  t )  <_  B } )
40 rfcnpre4.4 . . 3  |-  A  =  { t  e.  T  |  ( F `  t )  <_  B }
4139, 40syl6eqr 2493 . 2  |-  ( ph  ->  ( `' F "
( -oo (,] B ) )  =  A )
42 iocmnfcld 20348 . . . . 5  |-  ( B  e.  RR  ->  ( -oo (,] B )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )
4310, 42syl 16 . . . 4  |-  ( ph  ->  ( -oo (,] B
)  e.  ( Clsd `  ( topGen `  ran  (,) )
) )
441fveq2i 5694 . . . 4  |-  ( Clsd `  K )  =  (
Clsd `  ( topGen ` 
ran  (,) ) )
4543, 44syl6eleqr 2534 . . 3  |-  ( ph  ->  ( -oo (,] B
)  e.  ( Clsd `  K ) )
46 cnclima 18872 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( -oo (,] B )  e.  ( Clsd `  K
) )  ->  ( `' F " ( -oo (,] B ) )  e.  ( Clsd `  J
) )
474, 45, 46syl2anc 661 . 2  |-  ( ph  ->  ( `' F "
( -oo (,] B ) )  e.  ( Clsd `  J ) )
4841, 47eqeltrrd 2518 1  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   F/_wnfc 2566   {crab 2719   U.cuni 4091   class class class wbr 4292   `'ccnv 4839   ran crn 4841   "cima 4843    Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091   RRcr 9281   -oocmnf 9416   RR*cxr 9417    < clt 9418    <_ cle 9419   (,)cioo 11300   (,]cioc 11301   topGenctg 14376   Clsdccld 18620    Cn ccn 18828
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-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-ioo 11304  df-ioc 11305  df-topgen 14382  df-top 18503  df-bases 18505  df-topon 18506  df-cld 18623  df-cn 18831
This theorem is referenced by:  stoweidlem59  29854
  Copyright terms: Public domain W3C validator