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Theorem rfcnpre4 31214
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 2467 . . . . . . . 8  |-  ( J  Cn  K )  =  ( J  Cn  K
)
4 rfcnpre4.6 . . . . . . . 8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
51, 2, 3, 4fcnre 31205 . . . . . . 7  |-  ( ph  ->  F : T --> RR )
6 ffn 5731 . . . . . . 7  |-  ( F : T --> RR  ->  F  Fn  T )
7 elpreima 6002 . . . . . . 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 11324 . . . . . . . . 9  |- -oo  e.  RR*
10 rfcnpre4.5 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR )
1110rexrd 9644 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR* )
1211adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  T )  ->  B  e.  RR* )
13 elioc1 11572 . . . . . . . . 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 1004 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  T )  /\  (
( F `  s
)  e.  RR*  /\ -oo  <  ( F `  s
)  /\  ( F `  s )  <_  B
) )  ->  ( F `  s )  <_  B )
165fnvinran 31194 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  T )  ->  ( F `  s )  e.  RR )
1716rexrd 9644 . . . . . . . . . . 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 11334 . . . . . . . . . . 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 1176 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  (
( F `  s
)  e.  RR*  /\ -oo  <  ( F `  s
)  /\  ( F `  s )  <_  B
) )
2415, 23impbida 830 . . . . . . . 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 2629 . . . . . 6  |-  F/_ t
s
29 nfcv 2629 . . . . . 6  |-  F/_ t T
30 rfcnpre4.1 . . . . . . . 8  |-  F/_ t F
3130, 28nffv 5873 . . . . . . 7  |-  F/_ t
( F `  s
)
32 nfcv 2629 . . . . . . 7  |-  F/_ t  <_
33 nfcv 2629 . . . . . . 7  |-  F/_ t B
3431, 32, 33nfbr 4491 . . . . . 6  |-  F/ t ( F `  s
)  <_  B
35 fveq2 5866 . . . . . . 7  |-  ( t  =  s  ->  ( F `  t )  =  ( F `  s ) )
3635breq1d 4457 . . . . . 6  |-  ( t  =  s  ->  (
( F `  t
)  <_  B  <->  ( F `  s )  <_  B
) )
3728, 29, 34, 36elrabf 3259 . . . . 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 2464 . . 3  |-  ( ph  ->  ( `' F "
( -oo (,] B ) )  =  { t  e.  T  |  ( F `  t )  <_  B } )
40 rfcnpre4.4 . . 3  |-  A  =  { t  e.  T  |  ( F `  t )  <_  B }
4139, 40syl6eqr 2526 . 2  |-  ( ph  ->  ( `' F "
( -oo (,] B ) )  =  A )
42 iocmnfcld 21103 . . . . 5  |-  ( B  e.  RR  ->  ( -oo (,] B )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )
4310, 42syl 16 . . . 4  |-  ( ph  ->  ( -oo (,] B
)  e.  ( Clsd `  ( topGen `  ran  (,) )
) )
441fveq2i 5869 . . . 4  |-  ( Clsd `  K )  =  (
Clsd `  ( topGen ` 
ran  (,) ) )
4543, 44syl6eleqr 2566 . . 3  |-  ( ph  ->  ( -oo (,] B
)  e.  ( Clsd `  K ) )
46 cnclima 19575 . . 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 2556 1  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   F/_wnfc 2615   {crab 2818   U.cuni 4245   class class class wbr 4447   `'ccnv 4998   ran crn 5000   "cima 5002    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6285   RRcr 9492   -oocmnf 9627   RR*cxr 9628    < clt 9629    <_ cle 9630   (,)cioo 11530   (,]cioc 11531   topGenctg 14696   Clsdccld 19323    Cn ccn 19531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084  df-q 11184  df-ioo 11534  df-ioc 11535  df-topgen 14702  df-top 19206  df-bases 19208  df-topon 19209  df-cld 19326  df-cn 19534
This theorem is referenced by:  stoweidlem59  31586
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