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Theorem rfcnpre3 29760
Description: If F is a continuous function with respect to the standard topology, then the preimage A of the values greater or equal than a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
rfcnpre3.2  |-  F/_ t F
rfcnpre3.3  |-  K  =  ( topGen `  ran  (,) )
rfcnpre3.4  |-  T  = 
U. J
rfcnpre3.5  |-  A  =  { t  e.  T  |  B  <_  ( F `
 t ) }
rfcnpre3.6  |-  ( ph  ->  B  e.  RR )
rfcnpre3.8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
Assertion
Ref Expression
rfcnpre3  |-  ( 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 rfcnpre3
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 rfcnpre3.3 . . . . . . . 8  |-  K  =  ( topGen `  ran  (,) )
2 rfcnpre3.4 . . . . . . . 8  |-  T  = 
U. J
3 eqid 2443 . . . . . . . 8  |-  ( J  Cn  K )  =  ( J  Cn  K
)
4 rfcnpre3.8 . . . . . . . 8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
51, 2, 3, 4fcnre 29752 . . . . . . 7  |-  ( ph  ->  F : T --> RR )
6 ffn 5564 . . . . . . 7  |-  ( F : T --> RR  ->  F  Fn  T )
7 elpreima 5828 . . . . . . 7  |-  ( F  Fn  T  ->  (
s  e.  ( `' F " ( B [,) +oo ) )  <-> 
( s  e.  T  /\  ( F `  s
)  e.  ( B [,) +oo ) ) ) )
85, 6, 73syl 20 . . . . . 6  |-  ( ph  ->  ( s  e.  ( `' F " ( B [,) +oo ) )  <-> 
( s  e.  T  /\  ( F `  s
)  e.  ( B [,) +oo ) ) ) )
9 rfcnpre3.6 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR )
109rexrd 9438 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR* )
1110adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  T )  ->  B  e.  RR* )
12 pnfxr 11097 . . . . . . . . 9  |- +oo  e.  RR*
13 elico1 11348 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\ +oo  e.  RR* )  ->  (
( F `  s
)  e.  ( B [,) +oo )  <->  ( ( F `  s )  e.  RR*  /\  B  <_ 
( F `  s
)  /\  ( F `  s )  < +oo ) ) )
1411, 12, 13sylancl 662 . . . . . . . 8  |-  ( (
ph  /\  s  e.  T )  ->  (
( F `  s
)  e.  ( B [,) +oo )  <->  ( ( F `  s )  e.  RR*  /\  B  <_ 
( F `  s
)  /\  ( F `  s )  < +oo ) ) )
15 simpr2 995 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  T )  /\  (
( F `  s
)  e.  RR*  /\  B  <_  ( F `  s
)  /\  ( F `  s )  < +oo ) )  ->  B  <_  ( F `  s
) )
165fnvinran 29741 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  T )  ->  ( F `  s )  e.  RR )
1716rexrd 9438 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  T )  ->  ( F `  s )  e.  RR* )
1817adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  T )  /\  B  <_  ( F `  s
) )  ->  ( F `  s )  e.  RR* )
19 simpr 461 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  T )  /\  B  <_  ( F `  s
) )  ->  B  <_  ( F `  s
) )
2016adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  T )  /\  B  <_  ( F `  s
) )  ->  ( F `  s )  e.  RR )
21 ltpnf 11107 . . . . . . . . . . 11  |-  ( ( F `  s )  e.  RR  ->  ( F `  s )  < +oo )
2220, 21syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  T )  /\  B  <_  ( F `  s
) )  ->  ( F `  s )  < +oo )
2318, 19, 223jca 1168 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  T )  /\  B  <_  ( F `  s
) )  ->  (
( F `  s
)  e.  RR*  /\  B  <_  ( F `  s
)  /\  ( F `  s )  < +oo ) )
2415, 23impbida 828 . . . . . . . 8  |-  ( (
ph  /\  s  e.  T )  ->  (
( ( F `  s )  e.  RR*  /\  B  <_  ( F `  s )  /\  ( F `  s )  < +oo )  <->  B  <_  ( F `  s ) ) )
2514, 24bitrd 253 . . . . . . 7  |-  ( (
ph  /\  s  e.  T )  ->  (
( F `  s
)  e.  ( B [,) +oo )  <->  B  <_  ( F `  s ) ) )
2625pm5.32da 641 . . . . . 6  |-  ( ph  ->  ( ( s  e.  T  /\  ( F `
 s )  e.  ( B [,) +oo ) )  <->  ( s  e.  T  /\  B  <_ 
( F `  s
) ) ) )
278, 26bitrd 253 . . . . 5  |-  ( ph  ->  ( s  e.  ( `' F " ( B [,) +oo ) )  <-> 
( s  e.  T  /\  B  <_  ( F `
 s ) ) ) )
28 nfcv 2584 . . . . . 6  |-  F/_ t
s
29 nfcv 2584 . . . . . 6  |-  F/_ t T
30 nfcv 2584 . . . . . . 7  |-  F/_ t B
31 nfcv 2584 . . . . . . 7  |-  F/_ t  <_
32 rfcnpre3.2 . . . . . . . 8  |-  F/_ t F
3332, 28nffv 5703 . . . . . . 7  |-  F/_ t
( F `  s
)
3430, 31, 33nfbr 4341 . . . . . 6  |-  F/ t  B  <_  ( F `  s )
35 fveq2 5696 . . . . . . 7  |-  ( t  =  s  ->  ( F `  t )  =  ( F `  s ) )
3635breq2d 4309 . . . . . 6  |-  ( t  =  s  ->  ( B  <_  ( F `  t )  <->  B  <_  ( F `  s ) ) )
3728, 29, 34, 36elrabf 3120 . . . . 5  |-  ( s  e.  { t  e.  T  |  B  <_ 
( F `  t
) }  <->  ( s  e.  T  /\  B  <_ 
( F `  s
) ) )
3827, 37syl6bbr 263 . . . 4  |-  ( ph  ->  ( s  e.  ( `' F " ( B [,) +oo ) )  <-> 
s  e.  { t  e.  T  |  B  <_  ( F `  t
) } ) )
3938eqrdv 2441 . . 3  |-  ( ph  ->  ( `' F "
( B [,) +oo ) )  =  {
t  e.  T  |  B  <_  ( F `  t ) } )
40 rfcnpre3.5 . . 3  |-  A  =  { t  e.  T  |  B  <_  ( F `
 t ) }
4139, 40syl6eqr 2493 . 2  |-  ( ph  ->  ( `' F "
( B [,) +oo ) )  =  A )
42 icopnfcld 20352 . . . . 5  |-  ( B  e.  RR  ->  ( B [,) +oo )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )
439, 42syl 16 . . . 4  |-  ( ph  ->  ( B [,) +oo )  e.  ( Clsd `  ( topGen `  ran  (,) )
) )
441fveq2i 5699 . . . 4  |-  ( Clsd `  K )  =  (
Clsd `  ( topGen ` 
ran  (,) ) )
4543, 44syl6eleqr 2534 . . 3  |-  ( ph  ->  ( B [,) +oo )  e.  ( Clsd `  K ) )
46 cnclima 18877 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( B [,) +oo )  e.  ( Clsd `  K
) )  ->  ( `' F " ( B [,) +oo ) )  e.  ( Clsd `  J
) )
474, 45, 46syl2anc 661 . 2  |-  ( ph  ->  ( `' F "
( B [,) +oo ) )  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 2571   {crab 2724   U.cuni 4096   class class class wbr 4297   `'ccnv 4844   ran crn 4846   "cima 4848    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   RRcr 9286   +oocpnf 9420   RR*cxr 9422    < clt 9423    <_ cle 9424   (,)cioo 11305   [,)cico 11307   topGenctg 14381   Clsdccld 18625    Cn ccn 18833
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 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-ioo 11309  df-ico 11311  df-topgen 14387  df-top 18508  df-bases 18510  df-topon 18511  df-cld 18628  df-cn 18836
This theorem is referenced by:  stoweidlem59  29859
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