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Theorem rfcnpre2 27805
Description: If  F is a continuous function with respect to the standard topology, then the preimage A of the values smaller than a given extended real  B, is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
rfcnpre2.1  |-  F/_ x B
rfcnpre2.2  |-  F/_ x F
rfcnpre2.3  |-  F/ x ph
rfcnpre2.4  |-  K  =  ( topGen `  ran  (,) )
rfcnpre2.5  |-  X  = 
U. J
rfcnpre2.6  |-  A  =  { x  e.  X  |  ( F `  x )  <  B }
rfcnpre2.7  |-  ( ph  ->  B  e.  RR* )
rfcnpre2.8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
Assertion
Ref Expression
rfcnpre2  |-  ( ph  ->  A  e.  J )

Proof of Theorem rfcnpre2
StepHypRef Expression
1 rfcnpre2.3 . . . . 5  |-  F/ x ph
2 simpl 443 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ph )
3 rfcnpre2.4 . . . . . . . . . . . . 13  |-  K  =  ( topGen `  ran  (,) )
4 rfcnpre2.5 . . . . . . . . . . . . 13  |-  X  = 
U. J
5 eqid 2296 . . . . . . . . . . . . 13  |-  ( J  Cn  K )  =  ( J  Cn  K
)
6 rfcnpre2.8 . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
73, 4, 5, 6fcnre 27799 . . . . . . . . . . . 12  |-  ( ph  ->  F : X --> RR )
87adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  F : X --> RR )
9 simpr 447 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
108, 9jca 518 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  ( F : X --> RR  /\  x  e.  X )
)
11 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( F : X --> RR  /\  x  e.  X )  ->  ( F `  x
)  e.  RR )
1210, 11syl 15 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  RR )
132, 12jca 518 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  ( ph  /\  ( F `  x )  e.  RR ) )
14 rfcnpre2.7 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR* )
15 elioomnf 10754 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  ( ( F `  x )  e.  (  -oo (,) B )  <->  ( ( F `  x )  e.  RR  /\  ( F `
 x )  < 
B ) ) )
1614, 15syl 15 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  x )  e.  ( 
-oo (,) B )  <->  ( ( F `  x )  e.  RR  /\  ( F `
 x )  < 
B ) ) )
1716baibd 875 . . . . . . . 8  |-  ( (
ph  /\  ( F `  x )  e.  RR )  ->  ( ( F `
 x )  e.  (  -oo (,) B
)  <->  ( F `  x )  <  B
) )
1813, 17syl 15 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( F `  x
)  e.  (  -oo (,) B )  <->  ( F `  x )  <  B
) )
1918pm5.32da 622 . . . . . 6  |-  ( ph  ->  ( ( x  e.  X  /\  ( F `
 x )  e.  (  -oo (,) B
) )  <->  ( x  e.  X  /\  ( F `  x )  <  B ) ) )
20 ffn 5405 . . . . . . . 8  |-  ( F : X --> RR  ->  F  Fn  X )
217, 20syl 15 . . . . . . 7  |-  ( ph  ->  F  Fn  X )
22 elpreima 5661 . . . . . . 7  |-  ( F  Fn  X  ->  (
x  e.  ( `' F " (  -oo (,) B ) )  <->  ( x  e.  X  /\  ( F `  x )  e.  (  -oo (,) B
) ) ) )
2321, 22syl 15 . . . . . 6  |-  ( ph  ->  ( x  e.  ( `' F " (  -oo (,) B ) )  <->  ( x  e.  X  /\  ( F `  x )  e.  (  -oo (,) B
) ) ) )
24 rabid 2729 . . . . . . 7  |-  ( x  e.  { x  e.  X  |  ( F `
 x )  < 
B }  <->  ( x  e.  X  /\  ( F `  x )  <  B ) )
2524a1i 10 . . . . . 6  |-  ( ph  ->  ( x  e.  {
x  e.  X  | 
( F `  x
)  <  B }  <->  ( x  e.  X  /\  ( F `  x )  <  B ) ) )
2619, 23, 253bitr4d 276 . . . . 5  |-  ( ph  ->  ( x  e.  ( `' F " (  -oo (,) B ) )  <->  x  e.  { x  e.  X  | 
( F `  x
)  <  B }
) )
271, 26alrimi 1757 . . . 4  |-  ( ph  ->  A. x ( x  e.  ( `' F " (  -oo (,) B
) )  <->  x  e.  { x  e.  X  | 
( F `  x
)  <  B }
) )
28 rfcnpre2.2 . . . . . . 7  |-  F/_ x F
2928nfcnv 4876 . . . . . 6  |-  F/_ x `' F
30 nfcv 2432 . . . . . . 7  |-  F/_ x  -oo
31 nfcv 2432 . . . . . . 7  |-  F/_ x (,)
32 rfcnpre2.1 . . . . . . 7  |-  F/_ x B
3330, 31, 32nfov 5897 . . . . . 6  |-  F/_ x
(  -oo (,) B )
3429, 33nfima 5036 . . . . 5  |-  F/_ x
( `' F "
(  -oo (,) B ) )
35 nfrab1 2733 . . . . 5  |-  F/_ x { x  e.  X  |  ( F `  x )  <  B }
3634, 35cleqf 2456 . . . 4  |-  ( ( `' F " (  -oo (,) B ) )  =  { x  e.  X  |  ( F `  x )  <  B } 
<-> 
A. x ( x  e.  ( `' F " (  -oo (,) B
) )  <->  x  e.  { x  e.  X  | 
( F `  x
)  <  B }
) )
3727, 36sylibr 203 . . 3  |-  ( ph  ->  ( `' F "
(  -oo (,) B ) )  =  { x  e.  X  |  ( F `  x )  <  B } )
38 rfcnpre2.6 . . 3  |-  A  =  { x  e.  X  |  ( F `  x )  <  B }
3937, 38syl6eqr 2346 . 2  |-  ( ph  ->  ( `' F "
(  -oo (,) B ) )  =  A )
40 iooretop 18291 . . . . . 6  |-  (  -oo (,) B )  e.  (
topGen `  ran  (,) )
4140a1i 10 . . . . 5  |-  ( ph  ->  (  -oo (,) B
)  e.  ( topGen ` 
ran  (,) ) )
4241, 3syl6eleqr 2387 . . . 4  |-  ( ph  ->  (  -oo (,) B
)  e.  K )
436, 42jca 518 . . 3  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  /\  (  -oo (,) B )  e.  K
) )
44 cnima 17010 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  (  -oo (,) B )  e.  K )  -> 
( `' F "
(  -oo (,) B ) )  e.  J )
4543, 44syl 15 . 2  |-  ( ph  ->  ( `' F "
(  -oo (,) B ) )  e.  J )
4639, 45eqeltrrd 2371 1  |-  ( ph  ->  A  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   F/wnf 1534    = wceq 1632    e. wcel 1696   F/_wnfc 2419   {crab 2560   U.cuni 3843   class class class wbr 4039   `'ccnv 4704   ran crn 4706   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752    -oocmnf 8881   RR*cxr 8882    < clt 8883   (,)cioo 10672   topGenctg 13358    Cn ccn 16970
This theorem is referenced by:  stoweidlem52  27904
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-ioo 10676  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-cn 16973
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