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Theorem rfcnpre4 27705
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 rfcnpre4.6 . . . 4  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
2 rfcnpre4.5 . . . . . 6  |-  ( ph  ->  B  e.  RR )
3 iocmnfcld 18278 . . . . . 6  |-  ( B  e.  RR  ->  (  -oo (,] B )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )
42, 3syl 15 . . . . 5  |-  ( ph  ->  (  -oo (,] B
)  e.  ( Clsd `  ( topGen `  ran  (,) )
) )
5 rfcnpre4.2 . . . . . . . 8  |-  K  =  ( topGen `  ran  (,) )
65fveq2i 5528 . . . . . . 7  |-  ( Clsd `  K )  =  (
Clsd `  ( topGen ` 
ran  (,) ) )
76a1i 10 . . . . . 6  |-  ( ph  ->  ( Clsd `  K
)  =  ( Clsd `  ( topGen `  ran  (,) )
) )
87eleq2d 2350 . . . . 5  |-  ( ph  ->  ( (  -oo (,] B )  e.  (
Clsd `  K )  <->  ( 
-oo (,] B )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) ) )
94, 8mpbird 223 . . . 4  |-  ( ph  ->  (  -oo (,] B
)  e.  ( Clsd `  K ) )
101, 9jca 518 . . 3  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  /\  (  -oo (,] B )  e.  (
Clsd `  K )
) )
11 cnclima 16997 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  (  -oo (,] B )  e.  ( Clsd `  K
) )  ->  ( `' F " (  -oo (,] B ) )  e.  ( Clsd `  J
) )
1210, 11syl 15 . 2  |-  ( ph  ->  ( `' F "
(  -oo (,] B ) )  e.  ( Clsd `  J ) )
13 rfcnpre4.3 . . . . . . . . . 10  |-  T  = 
U. J
14 eqid 2283 . . . . . . . . . 10  |-  ( J  Cn  K )  =  ( J  Cn  K
)
155, 13, 14, 1fcnre 27696 . . . . . . . . 9  |-  ( ph  ->  F : T --> RR )
16 ffn 5389 . . . . . . . . 9  |-  ( F : T --> RR  ->  F  Fn  T )
1715, 16syl 15 . . . . . . . 8  |-  ( ph  ->  F  Fn  T )
18 elpreima 5645 . . . . . . . 8  |-  ( F  Fn  T  ->  (
s  e.  ( `' F " (  -oo (,] B ) )  <->  ( s  e.  T  /\  ( F `  s )  e.  (  -oo (,] B
) ) ) )
1917, 18syl 15 . . . . . . 7  |-  ( ph  ->  ( s  e.  ( `' F " (  -oo (,] B ) )  <->  ( s  e.  T  /\  ( F `  s )  e.  (  -oo (,] B
) ) ) )
20 mnfxr 10456 . . . . . . . . . . . 12  |-  -oo  e.  RR*
2120a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  T )  ->  -oo  e.  RR* )
22 rexr 8877 . . . . . . . . . . . . 13  |-  ( B  e.  RR  ->  B  e.  RR* )
232, 22syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  RR* )
2423adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  T )  ->  B  e.  RR* )
2521, 24jca 518 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  T )  ->  (  -oo  e.  RR*  /\  B  e. 
RR* ) )
26 elioc1 10698 . . . . . . . . . 10  |-  ( ( 
-oo  e.  RR*  /\  B  e.  RR* )  ->  (
( F `  s
)  e.  (  -oo (,] B )  <->  ( ( F `  s )  e.  RR*  /\  -oo  <  ( F `  s )  /\  ( F `  s )  <_  B
) ) )
2725, 26syl 15 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  T )  ->  (
( F `  s
)  e.  (  -oo (,] B )  <->  ( ( F `  s )  e.  RR*  /\  -oo  <  ( F `  s )  /\  ( F `  s )  <_  B
) ) )
28 simpr3 963 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  T )  /\  (
( F `  s
)  e.  RR*  /\  -oo  <  ( F `  s
)  /\  ( F `  s )  <_  B
) )  ->  ( F `  s )  <_  B )
2915adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  s  e.  T )  ->  F : T --> RR )
30 simpr 447 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  s  e.  T )  ->  s  e.  T )
3129, 30jca 518 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  s  e.  T )  ->  ( F : T --> RR  /\  s  e.  T )
)
32 ffvelrn 5663 . . . . . . . . . . . . . 14  |-  ( ( F : T --> RR  /\  s  e.  T )  ->  ( F `  s
)  e.  RR )
3331, 32syl 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\  s  e.  T )  ->  ( F `  s )  e.  RR )
34 rexr 8877 . . . . . . . . . . . . 13  |-  ( ( F `  s )  e.  RR  ->  ( F `  s )  e.  RR* )
3533, 34syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  T )  ->  ( F `  s )  e.  RR* )
3635adantr 451 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  ( F `  s )  e.  RR* )
3733adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  ( F `  s )  e.  RR )
38 mnflt 10464 . . . . . . . . . . . 12  |-  ( ( F `  s )  e.  RR  ->  -oo  <  ( F `  s ) )
3937, 38syl 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  -oo  <  ( F `  s ) )
40 simpr 447 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  ( F `  s )  <_  B )
4136, 39, 403jca 1132 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  (
( F `  s
)  e.  RR*  /\  -oo  <  ( F `  s
)  /\  ( F `  s )  <_  B
) )
4228, 41impbida 805 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  T )  ->  (
( ( F `  s )  e.  RR*  /\ 
-oo  <  ( F `  s )  /\  ( F `  s )  <_  B )  <->  ( F `  s )  <_  B
) )
4327, 42bitrd 244 . . . . . . . 8  |-  ( (
ph  /\  s  e.  T )  ->  (
( F `  s
)  e.  (  -oo (,] B )  <->  ( F `  s )  <_  B
) )
4443pm5.32da 622 . . . . . . 7  |-  ( ph  ->  ( ( s  e.  T  /\  ( F `
 s )  e.  (  -oo (,] B
) )  <->  ( s  e.  T  /\  ( F `  s )  <_  B ) ) )
4519, 44bitrd 244 . . . . . 6  |-  ( ph  ->  ( s  e.  ( `' F " (  -oo (,] B ) )  <->  ( s  e.  T  /\  ( F `  s )  <_  B ) ) )
46 nfcv 2419 . . . . . . 7  |-  F/_ t
s
47 nfcv 2419 . . . . . . 7  |-  F/_ t T
48 rfcnpre4.1 . . . . . . . . 9  |-  F/_ t F
4948, 46nffv 5532 . . . . . . . 8  |-  F/_ t
( F `  s
)
50 nfcv 2419 . . . . . . . 8  |-  F/_ t  <_
51 nfcv 2419 . . . . . . . 8  |-  F/_ t B
5249, 50, 51nfbr 4067 . . . . . . 7  |-  F/ t ( F `  s
)  <_  B
53 fveq2 5525 . . . . . . . 8  |-  ( t  =  s  ->  ( F `  t )  =  ( F `  s ) )
5453breq1d 4033 . . . . . . 7  |-  ( t  =  s  ->  (
( F `  t
)  <_  B  <->  ( F `  s )  <_  B
) )
5546, 47, 52, 54elrabf 2922 . . . . . 6  |-  ( s  e.  { t  e.  T  |  ( F `
 t )  <_  B }  <->  ( s  e.  T  /\  ( F `
 s )  <_  B ) )
5645, 55syl6bbr 254 . . . . 5  |-  ( ph  ->  ( s  e.  ( `' F " (  -oo (,] B ) )  <->  s  e.  { t  e.  T  | 
( F `  t
)  <_  B }
) )
5756eqrdv 2281 . . . 4  |-  ( ph  ->  ( `' F "
(  -oo (,] B ) )  =  { t  e.  T  |  ( F `  t )  <_  B } )
58 rfcnpre4.4 . . . 4  |-  A  =  { t  e.  T  |  ( F `  t )  <_  B }
5957, 58syl6eqr 2333 . . 3  |-  ( ph  ->  ( `' F "
(  -oo (,] B ) )  =  A )
6059eleq1d 2349 . 2  |-  ( ph  ->  ( ( `' F " (  -oo (,] B
) )  e.  (
Clsd `  J )  <->  A  e.  ( Clsd `  J
) ) )
6112, 60mpbid 201 1  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   F/_wnfc 2406   {crab 2547   U.cuni 3827   class class class wbr 4023   `'ccnv 4688   ran crn 4690   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868   (,)cioo 10656   (,]cioc 10657   topGenctg 13342   Clsdccld 16753    Cn ccn 16954
This theorem is referenced by:  stoweidlem59  27808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-ioo 10660  df-ioc 10661  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-cn 16957
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