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Theorem rfcnpre1 30972
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 19504 . . . . . . . . . . . . . . 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 538 . . . . . . . . . . . . 13  |-  ( ph  ->  ( J  e.  Top  /\  X  =  U. J
) )
7 istopon 19190 . . . . . . . . . . . . 13  |-  ( J  e.  (TopOn `  X
)  <->  ( J  e. 
Top  /\  X  =  U. J ) )
86, 7sylibr 212 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  (TopOn `  X ) )
9 rfcnpre1.4 . . . . . . . . . . . . 13  |-  K  =  ( topGen `  ran  (,) )
10 retopon 21002 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
119, 10eqeltri 2551 . . . . . . . . . . . 12  |-  K  e.  (TopOn `  RR )
12 iscn 19499 . . . . . . . . . . . 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 662 . . . . . . . . . . 11  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
( F : X --> RR  /\  A. y  e.  K  ( `' F " y )  e.  J
) ) )
142, 13mpbid 210 . . . . . . . . . 10  |-  ( ph  ->  ( F : X --> RR  /\  A. y  e.  K  ( `' F " y )  e.  J
) )
1514simpld 459 . . . . . . . . 9  |-  ( ph  ->  F : X --> RR )
1615fnvinran 30967 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  RR )
17 rfcnpre1.7 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR* )
18 elioopnf 11614 . . . . . . . . . 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 907 . . . . . . . 8  |-  ( (
ph  /\  ( F `  x )  e.  RR )  ->  ( ( F `
 x )  e.  ( B (,) +oo ) 
<->  B  <  ( F `
 x ) ) )
2116, 20syldan 470 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( F `  x
)  e.  ( B (,) +oo )  <->  B  <  ( F `  x ) ) )
2221pm5.32da 641 . . . . . 6  |-  ( ph  ->  ( ( x  e.  X  /\  ( F `
 x )  e.  ( B (,) +oo ) )  <->  ( x  e.  X  /\  B  < 
( F `  x
) ) ) )
23 ffn 5729 . . . . . . 7  |-  ( F : X --> RR  ->  F  Fn  X )
24 elpreima 5999 . . . . . . 7  |-  ( F  Fn  X  ->  (
x  e.  ( `' F " ( B (,) +oo ) )  <-> 
( x  e.  X  /\  ( F `  x
)  e.  ( B (,) +oo ) ) ) )
2515, 23, 243syl 20 . . . . . 6  |-  ( ph  ->  ( x  e.  ( `' F " ( B (,) +oo ) )  <-> 
( x  e.  X  /\  ( F `  x
)  e.  ( B (,) +oo ) ) ) )
26 rabid 3038 . . . . . . 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 285 . . . . 5  |-  ( ph  ->  ( x  e.  ( `' F " ( B (,) +oo ) )  <-> 
x  e.  { x  e.  X  |  B  <  ( F `  x
) } ) )
291, 28alrimi 1825 . . . 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 5179 . . . . . 6  |-  F/_ x `' F
32 rfcnpre1.1 . . . . . . 7  |-  F/_ x B
33 nfcv 2629 . . . . . . 7  |-  F/_ x (,)
34 nfcv 2629 . . . . . . 7  |-  F/_ x +oo
3532, 33, 34nfov 6305 . . . . . 6  |-  F/_ x
( B (,) +oo )
3631, 35nfima 5343 . . . . 5  |-  F/_ x
( `' F "
( B (,) +oo ) )
37 nfrab1 3042 . . . . 5  |-  F/_ x { x  e.  X  |  B  <  ( F `
 x ) }
3836, 37cleqf 2656 . . . 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 212 . . 3  |-  ( ph  ->  ( `' F "
( B (,) +oo ) )  =  {
x  e.  X  |  B  <  ( F `  x ) } )
40 rfcnpre1.6 . . 3  |-  A  =  { x  e.  X  |  B  <  ( F `
 x ) }
4139, 40syl6eqr 2526 . 2  |-  ( ph  ->  ( `' F "
( B (,) +oo ) )  =  A )
42 iooretop 21005 . . . 4  |-  ( B (,) +oo )  e.  ( topGen `  ran  (,) )
4342, 9eleqtrri 2554 . . 3  |-  ( B (,) +oo )  e.  K
44 cnima 19529 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( B (,) +oo )  e.  K )  ->  ( `' F " ( B (,) +oo ) )  e.  J )
452, 43, 44sylancl 662 . 2  |-  ( ph  ->  ( `' F "
( B (,) +oo ) )  e.  J
)
4641, 45eqeltrrd 2556 1  |-  ( ph  ->  A  e.  J )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377    = wceq 1379   F/wnf 1599    e. wcel 1767   F/_wnfc 2615   A.wral 2814   {crab 2818   U.cuni 4245   class class class wbr 4447   `'ccnv 4998   ran crn 5000   "cima 5002    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   RRcr 9487   +oocpnf 9621   RR*cxr 9623    < clt 9624   (,)cioo 11525   topGenctg 14686   Topctop 19158  TopOnctopon 19159    Cn ccn 19488
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 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-ioo 11529  df-topgen 14692  df-top 19163  df-bases 19165  df-topon 19166  df-cn 19491
This theorem is referenced by:  stoweidlem46  31346
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