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Theorem rfcnpre4 27681
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 2436 . . . . . . . 8  |-  ( J  Cn  K )  =  ( J  Cn  K
)
4 rfcnpre4.6 . . . . . . . 8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
51, 2, 3, 4fcnre 27672 . . . . . . 7  |-  ( ph  ->  F : T --> RR )
6 ffn 5591 . . . . . . 7  |-  ( F : T --> RR  ->  F  Fn  T )
7 elpreima 5850 . . . . . . 7  |-  ( F  Fn  T  ->  (
s  e.  ( `' F " (  -oo (,] B ) )  <->  ( s  e.  T  /\  ( F `  s )  e.  (  -oo (,] B
) ) ) )
85, 6, 73syl 19 . . . . . 6  |-  ( ph  ->  ( s  e.  ( `' F " (  -oo (,] B ) )  <->  ( s  e.  T  /\  ( F `  s )  e.  (  -oo (,] B
) ) ) )
9 mnfxr 10714 . . . . . . . . 9  |-  -oo  e.  RR*
10 rfcnpre4.5 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR )
1110rexrd 9134 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR* )
1211adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  T )  ->  B  e.  RR* )
13 elioc1 10958 . . . . . . . . 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 645 . . . . . . . 8  |-  ( (
ph  /\  s  e.  T )  ->  (
( F `  s
)  e.  (  -oo (,] B )  <->  ( ( F `  s )  e.  RR*  /\  -oo  <  ( F `  s )  /\  ( F `  s )  <_  B
) ) )
15 simpr3 965 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  T )  /\  (
( F `  s
)  e.  RR*  /\  -oo  <  ( F `  s
)  /\  ( F `  s )  <_  B
) )  ->  ( F `  s )  <_  B )
165fnvinran 27661 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  T )  ->  ( F `  s )  e.  RR )
1716rexrd 9134 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  T )  ->  ( F `  s )  e.  RR* )
1817adantr 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  ( F `  s )  e.  RR* )
1916adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  ( F `  s )  e.  RR )
20 mnflt 10722 . . . . . . . . . . 11  |-  ( ( F `  s )  e.  RR  ->  -oo  <  ( F `  s ) )
2119, 20syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  -oo  <  ( F `  s ) )
22 simpr 448 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  ( F `  s )  <_  B )
2318, 21, 223jca 1134 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  (
( F `  s
)  e.  RR*  /\  -oo  <  ( F `  s
)  /\  ( F `  s )  <_  B
) )
2415, 23impbida 806 . . . . . . . 8  |-  ( (
ph  /\  s  e.  T )  ->  (
( ( F `  s )  e.  RR*  /\ 
-oo  <  ( F `  s )  /\  ( F `  s )  <_  B )  <->  ( F `  s )  <_  B
) )
2514, 24bitrd 245 . . . . . . 7  |-  ( (
ph  /\  s  e.  T )  ->  (
( F `  s
)  e.  (  -oo (,] B )  <->  ( F `  s )  <_  B
) )
2625pm5.32da 623 . . . . . 6  |-  ( ph  ->  ( ( s  e.  T  /\  ( F `
 s )  e.  (  -oo (,] B
) )  <->  ( s  e.  T  /\  ( F `  s )  <_  B ) ) )
278, 26bitrd 245 . . . . 5  |-  ( ph  ->  ( s  e.  ( `' F " (  -oo (,] B ) )  <->  ( s  e.  T  /\  ( F `  s )  <_  B ) ) )
28 nfcv 2572 . . . . . 6  |-  F/_ t
s
29 nfcv 2572 . . . . . 6  |-  F/_ t T
30 rfcnpre4.1 . . . . . . . 8  |-  F/_ t F
3130, 28nffv 5735 . . . . . . 7  |-  F/_ t
( F `  s
)
32 nfcv 2572 . . . . . . 7  |-  F/_ t  <_
33 nfcv 2572 . . . . . . 7  |-  F/_ t B
3431, 32, 33nfbr 4256 . . . . . 6  |-  F/ t ( F `  s
)  <_  B
35 fveq2 5728 . . . . . . 7  |-  ( t  =  s  ->  ( F `  t )  =  ( F `  s ) )
3635breq1d 4222 . . . . . 6  |-  ( t  =  s  ->  (
( F `  t
)  <_  B  <->  ( F `  s )  <_  B
) )
3728, 29, 34, 36elrabf 3091 . . . . 5  |-  ( s  e.  { t  e.  T  |  ( F `
 t )  <_  B }  <->  ( s  e.  T  /\  ( F `
 s )  <_  B ) )
3827, 37syl6bbr 255 . . . 4  |-  ( ph  ->  ( s  e.  ( `' F " (  -oo (,] B ) )  <->  s  e.  { t  e.  T  | 
( F `  t
)  <_  B }
) )
3938eqrdv 2434 . . 3  |-  ( ph  ->  ( `' F "
(  -oo (,] B ) )  =  { t  e.  T  |  ( F `  t )  <_  B } )
40 rfcnpre4.4 . . 3  |-  A  =  { t  e.  T  |  ( F `  t )  <_  B }
4139, 40syl6eqr 2486 . 2  |-  ( ph  ->  ( `' F "
(  -oo (,] B ) )  =  A )
42 iocmnfcld 18803 . . . . 5  |-  ( B  e.  RR  ->  (  -oo (,] B )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )
4310, 42syl 16 . . . 4  |-  ( ph  ->  (  -oo (,] B
)  e.  ( Clsd `  ( topGen `  ran  (,) )
) )
441fveq2i 5731 . . . 4  |-  ( Clsd `  K )  =  (
Clsd `  ( topGen ` 
ran  (,) ) )
4543, 44syl6eleqr 2527 . . 3  |-  ( ph  ->  (  -oo (,] B
)  e.  ( Clsd `  K ) )
46 cnclima 17332 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  (  -oo (,] B )  e.  ( Clsd `  K
) )  ->  ( `' F " (  -oo (,] B ) )  e.  ( Clsd `  J
) )
474, 45, 46syl2anc 643 . 2  |-  ( ph  ->  ( `' F "
(  -oo (,] B ) )  e.  ( Clsd `  J ) )
4841, 47eqeltrrd 2511 1  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   F/_wnfc 2559   {crab 2709   U.cuni 4015   class class class wbr 4212   `'ccnv 4877   ran crn 4879   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   RRcr 8989    -oocmnf 9118   RR*cxr 9119    < clt 9120    <_ cle 9121   (,)cioo 10916   (,]cioc 10917   topGenctg 13665   Clsdccld 17080    Cn ccn 17288
This theorem is referenced by:  stoweidlem59  27784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-ioo 10920  df-ioc 10921  df-topgen 13667  df-top 16963  df-bases 16965  df-topon 16966  df-cld 17083  df-cn 17291
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