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Theorem rfcnpre3 31569
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 2457 . . . . . . . 8  |-  ( J  Cn  K )  =  ( J  Cn  K
)
4 rfcnpre3.8 . . . . . . . 8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
51, 2, 3, 4fcnre 31561 . . . . . . 7  |-  ( ph  ->  F : T --> RR )
6 ffn 5737 . . . . . . 7  |-  ( F : T --> RR  ->  F  Fn  T )
7 elpreima 6008 . . . . . . 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 9660 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR* )
1110adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  T )  ->  B  e.  RR* )
12 pnfxr 11346 . . . . . . . . 9  |- +oo  e.  RR*
13 elico1 11597 . . . . . . . . 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 1003 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  T )  /\  (
( F `  s
)  e.  RR*  /\  B  <_  ( F `  s
)  /\  ( F `  s )  < +oo ) )  ->  B  <_  ( F `  s
) )
165fnvinran 31550 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  T )  ->  ( F `  s )  e.  RR )
1716rexrd 9660 . . . . . . . . . . 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 11356 . . . . . . . . . . 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 1176 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  T )  /\  B  <_  ( F `  s
) )  ->  (
( F `  s
)  e.  RR*  /\  B  <_  ( F `  s
)  /\  ( F `  s )  < +oo ) )
2415, 23impbida 832 . . . . . . . 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 2619 . . . . . 6  |-  F/_ t
s
29 nfcv 2619 . . . . . 6  |-  F/_ t T
30 nfcv 2619 . . . . . . 7  |-  F/_ t B
31 nfcv 2619 . . . . . . 7  |-  F/_ t  <_
32 rfcnpre3.2 . . . . . . . 8  |-  F/_ t F
3332, 28nffv 5879 . . . . . . 7  |-  F/_ t
( F `  s
)
3430, 31, 33nfbr 4500 . . . . . 6  |-  F/ t  B  <_  ( F `  s )
35 fveq2 5872 . . . . . . 7  |-  ( t  =  s  ->  ( F `  t )  =  ( F `  s ) )
3635breq2d 4468 . . . . . 6  |-  ( t  =  s  ->  ( B  <_  ( F `  t )  <->  B  <_  ( F `  s ) ) )
3728, 29, 34, 36elrabf 3255 . . . . 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 2454 . . 3  |-  ( ph  ->  ( `' F "
( B [,) +oo ) )  =  {
t  e.  T  |  B  <_  ( F `  t ) } )
40 rfcnpre3.5 . . 3  |-  A  =  { t  e.  T  |  B  <_  ( F `
 t ) }
4139, 40syl6eqr 2516 . 2  |-  ( ph  ->  ( `' F "
( B [,) +oo ) )  =  A )
42 icopnfcld 21400 . . . . 5  |-  ( B  e.  RR  ->  ( B [,) +oo )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )
439, 42syl 16 . . . 4  |-  ( ph  ->  ( B [,) +oo )  e.  ( Clsd `  ( topGen `  ran  (,) )
) )
441fveq2i 5875 . . . 4  |-  ( Clsd `  K )  =  (
Clsd `  ( topGen ` 
ran  (,) ) )
4543, 44syl6eleqr 2556 . . 3  |-  ( ph  ->  ( B [,) +oo )  e.  ( Clsd `  K ) )
46 cnclima 19895 . . 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 2546 1  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   F/_wnfc 2605   {crab 2811   U.cuni 4251   class class class wbr 4456   `'ccnv 5007   ran crn 5009   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   RRcr 9508   +oocpnf 9642   RR*cxr 9644    < clt 9645    <_ cle 9646   (,)cioo 11554   [,)cico 11556   topGenctg 14854   Clsdccld 19643    Cn ccn 19851
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-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  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-iun 4334  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-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-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  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-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-ioo 11558  df-ico 11560  df-topgen 14860  df-top 19525  df-bases 19527  df-topon 19528  df-cld 19646  df-cn 19854
This theorem is referenced by:  stoweidlem59  32002
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