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Theorem rfcnpre1 27565
Description: If F is a continuous function with respect to the standard topology, then the preimage A of the values greater than a given extended real B is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
rfcnpre1.1  |-  F/_ x B
rfcnpre1.2  |-  F/_ x F
rfcnpre1.3  |-  F/ x ph
rfcnpre1.4  |-  K  =  ( topGen `  ran  (,) )
rfcnpre1.5  |-  X  = 
U. J
rfcnpre1.6  |-  A  =  { x  e.  X  |  B  <  ( F `
 x ) }
rfcnpre1.7  |-  ( ph  ->  B  e.  RR* )
rfcnpre1.8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
Assertion
Ref Expression
rfcnpre1  |-  ( ph  ->  A  e.  J )

Proof of Theorem rfcnpre1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 rfcnpre1.3 . . . . 5  |-  F/ x ph
2 rfcnpre1.8 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
3 cntop1 17266 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
42, 3syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  J  e.  Top )
5 rfcnpre1.5 . . . . . . . . . . . . . 14  |-  X  = 
U. J
64, 5jctir 525 . . . . . . . . . . . . 13  |-  ( ph  ->  ( J  e.  Top  /\  X  =  U. J
) )
7 istopon 16953 . . . . . . . . . . . . 13  |-  ( J  e.  (TopOn `  X
)  <->  ( J  e. 
Top  /\  X  =  U. J ) )
86, 7sylibr 204 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  (TopOn `  X ) )
9 rfcnpre1.4 . . . . . . . . . . . . 13  |-  K  =  ( topGen `  ran  (,) )
10 retopon 18758 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
119, 10eqeltri 2482 . . . . . . . . . . . 12  |-  K  e.  (TopOn `  RR )
12 iscn 17261 . . . . . . . . . . . 12  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> RR  /\  A. y  e.  K  ( `' F " y )  e.  J ) ) )
138, 11, 12sylancl 644 . . . . . . . . . . 11  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
( F : X --> RR  /\  A. y  e.  K  ( `' F " y )  e.  J
) ) )
142, 13mpbid 202 . . . . . . . . . 10  |-  ( ph  ->  ( F : X --> RR  /\  A. y  e.  K  ( `' F " y )  e.  J
) )
1514simpld 446 . . . . . . . . 9  |-  ( ph  ->  F : X --> RR )
1615fnvinran 27560 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  RR )
17 rfcnpre1.7 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR* )
18 elioopnf 10962 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  ( ( F `  x )  e.  ( B (,)  +oo )  <->  ( ( F `
 x )  e.  RR  /\  B  < 
( F `  x
) ) ) )
1917, 18syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  x )  e.  ( B (,)  +oo )  <->  ( ( F `  x
)  e.  RR  /\  B  <  ( F `  x ) ) ) )
2019baibd 876 . . . . . . . 8  |-  ( (
ph  /\  ( F `  x )  e.  RR )  ->  ( ( F `
 x )  e.  ( B (,)  +oo ) 
<->  B  <  ( F `
 x ) ) )
2116, 20syldan 457 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( F `  x
)  e.  ( B (,)  +oo )  <->  B  <  ( F `  x ) ) )
2221pm5.32da 623 . . . . . 6  |-  ( ph  ->  ( ( x  e.  X  /\  ( F `
 x )  e.  ( B (,)  +oo ) )  <->  ( x  e.  X  /\  B  < 
( F `  x
) ) ) )
23 ffn 5558 . . . . . . 7  |-  ( F : X --> RR  ->  F  Fn  X )
24 elpreima 5817 . . . . . . 7  |-  ( F  Fn  X  ->  (
x  e.  ( `' F " ( B (,)  +oo ) )  <->  ( x  e.  X  /\  ( F `  x )  e.  ( B (,)  +oo ) ) ) )
2515, 23, 243syl 19 . . . . . 6  |-  ( ph  ->  ( x  e.  ( `' F " ( B (,)  +oo ) )  <->  ( x  e.  X  /\  ( F `  x )  e.  ( B (,)  +oo ) ) ) )
26 rabid 2852 . . . . . . 7  |-  ( x  e.  { x  e.  X  |  B  < 
( F `  x
) }  <->  ( x  e.  X  /\  B  < 
( F `  x
) ) )
2726a1i 11 . . . . . 6  |-  ( ph  ->  ( x  e.  {
x  e.  X  |  B  <  ( F `  x ) }  <->  ( x  e.  X  /\  B  < 
( F `  x
) ) ) )
2822, 25, 273bitr4d 277 . . . . 5  |-  ( ph  ->  ( x  e.  ( `' F " ( B (,)  +oo ) )  <->  x  e.  { x  e.  X  |  B  <  ( F `  x ) } ) )
291, 28alrimi 1777 . . . 4  |-  ( ph  ->  A. x ( x  e.  ( `' F " ( B (,)  +oo ) )  <->  x  e.  { x  e.  X  |  B  <  ( F `  x ) } ) )
30 rfcnpre1.2 . . . . . . 7  |-  F/_ x F
3130nfcnv 5018 . . . . . 6  |-  F/_ x `' F
32 rfcnpre1.1 . . . . . . 7  |-  F/_ x B
33 nfcv 2548 . . . . . . 7  |-  F/_ x (,)
34 nfcv 2548 . . . . . . 7  |-  F/_ x  +oo
3532, 33, 34nfov 6071 . . . . . 6  |-  F/_ x
( B (,)  +oo )
3631, 35nfima 5178 . . . . 5  |-  F/_ x
( `' F "
( B (,)  +oo ) )
37 nfrab1 2856 . . . . 5  |-  F/_ x { x  e.  X  |  B  <  ( F `
 x ) }
3836, 37cleqf 2572 . . . 4  |-  ( ( `' F " ( B (,)  +oo ) )  =  { x  e.  X  |  B  <  ( F `
 x ) }  <->  A. x ( x  e.  ( `' F "
( B (,)  +oo ) )  <->  x  e.  { x  e.  X  |  B  <  ( F `  x ) } ) )
3929, 38sylibr 204 . . 3  |-  ( ph  ->  ( `' F "
( B (,)  +oo ) )  =  {
x  e.  X  |  B  <  ( F `  x ) } )
40 rfcnpre1.6 . . 3  |-  A  =  { x  e.  X  |  B  <  ( F `
 x ) }
4139, 40syl6eqr 2462 . 2  |-  ( ph  ->  ( `' F "
( B (,)  +oo ) )  =  A )
42 iooretop 18761 . . . 4  |-  ( B (,)  +oo )  e.  (
topGen `  ran  (,) )
4342, 9eleqtrri 2485 . . 3  |-  ( B (,)  +oo )  e.  K
44 cnima 17291 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( B (,)  +oo )  e.  K )  ->  ( `' F " ( B (,)  +oo ) )  e.  J )
452, 43, 44sylancl 644 . 2  |-  ( ph  ->  ( `' F "
( B (,)  +oo ) )  e.  J
)
4641, 45eqeltrrd 2487 1  |-  ( ph  ->  A  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   F/wnf 1550    = wceq 1649    e. wcel 1721   F/_wnfc 2535   A.wral 2674   {crab 2678   U.cuni 3983   class class class wbr 4180   `'ccnv 4844   ran crn 4846   "cima 4848    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6048   RRcr 8953    +oocpnf 9081   RR*cxr 9083    < clt 9084   (,)cioo 10880   topGenctg 13628   Topctop 16921  TopOnctopon 16922    Cn ccn 17250
This theorem is referenced by:  stoweidlem46  27670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-q 10539  df-ioo 10884  df-topgen 13630  df-top 16926  df-bases 16928  df-topon 16929  df-cn 17253
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