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Theorem rfcnpre1 27690
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 simpl 443 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ph )
3 rfcnpre1.8 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
4 cntop1 16970 . . . . . . . . . . . . . . . . . . 19  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
53, 4syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  J  e.  Top )
6 rfcnpre1.5 . . . . . . . . . . . . . . . . . . 19  |-  X  = 
U. J
76a1i 10 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  X  =  U. J
)
85, 7jca 518 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( J  e.  Top  /\  X  =  U. J
) )
9 istopon 16663 . . . . . . . . . . . . . . . . 17  |-  ( J  e.  (TopOn `  X
)  <->  ( J  e. 
Top  /\  X  =  U. J ) )
108, 9sylibr 203 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  J  e.  (TopOn `  X ) )
11 retopon 18272 . . . . . . . . . . . . . . . . . 18  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
12 rfcnpre1.4 . . . . . . . . . . . . . . . . . . 19  |-  K  =  ( topGen `  ran  (,) )
1312eleq1i 2346 . . . . . . . . . . . . . . . . . 18  |-  ( K  e.  (TopOn `  RR ) 
<->  ( topGen `  ran  (,) )  e.  (TopOn `  RR )
)
1411, 13mpbir 200 . . . . . . . . . . . . . . . . 17  |-  K  e.  (TopOn `  RR )
1514a1i 10 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  K  e.  (TopOn `  RR ) )
1610, 15jca 518 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )
) )
17 iscn 16965 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )
)  ->  ( F  e.  ( J  Cn  K
)  <->  ( F : X
--> RR  /\  A. y  e.  K  ( `' F " y )  e.  J ) ) )
1816, 17syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  <-> 
( F : X --> RR  /\  A. y  e.  K  ( `' F " y )  e.  J
) ) )
193, 18mpbid 201 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F : X --> RR  /\  A. y  e.  K  ( `' F " y )  e.  J
) )
2019simpld 445 . . . . . . . . . . . 12  |-  ( ph  ->  F : X --> RR )
2120adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  F : X --> RR )
22 simpr 447 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
2321, 22jca 518 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  ( F : X --> RR  /\  x  e.  X )
)
24 ffvelrn 5663 . . . . . . . . . 10  |-  ( ( F : X --> RR  /\  x  e.  X )  ->  ( F `  x
)  e.  RR )
2523, 24syl 15 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  RR )
262, 25jca 518 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  ( ph  /\  ( F `  x )  e.  RR ) )
27 rfcnpre1.7 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR* )
28 elioopnf 10737 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  ( ( F `  x )  e.  ( B (,)  +oo )  <->  ( ( F `
 x )  e.  RR  /\  B  < 
( F `  x
) ) ) )
2927, 28syl 15 . . . . . . . . 9  |-  ( ph  ->  ( ( F `  x )  e.  ( B (,)  +oo )  <->  ( ( F `  x
)  e.  RR  /\  B  <  ( F `  x ) ) ) )
3029baibd 875 . . . . . . . 8  |-  ( (
ph  /\  ( F `  x )  e.  RR )  ->  ( ( F `
 x )  e.  ( B (,)  +oo ) 
<->  B  <  ( F `
 x ) ) )
3126, 30syl 15 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( F `  x
)  e.  ( B (,)  +oo )  <->  B  <  ( F `  x ) ) )
3231pm5.32da 622 . . . . . 6  |-  ( ph  ->  ( ( x  e.  X  /\  ( F `
 x )  e.  ( B (,)  +oo ) )  <->  ( x  e.  X  /\  B  < 
( F `  x
) ) ) )
33 ffn 5389 . . . . . . . 8  |-  ( F : X --> RR  ->  F  Fn  X )
3420, 33syl 15 . . . . . . 7  |-  ( ph  ->  F  Fn  X )
35 elpreima 5645 . . . . . . 7  |-  ( F  Fn  X  ->  (
x  e.  ( `' F " ( B (,)  +oo ) )  <->  ( x  e.  X  /\  ( F `  x )  e.  ( B (,)  +oo ) ) ) )
3634, 35syl 15 . . . . . 6  |-  ( ph  ->  ( x  e.  ( `' F " ( B (,)  +oo ) )  <->  ( x  e.  X  /\  ( F `  x )  e.  ( B (,)  +oo ) ) ) )
37 rabid 2716 . . . . . . 7  |-  ( x  e.  { x  e.  X  |  B  < 
( F `  x
) }  <->  ( x  e.  X  /\  B  < 
( F `  x
) ) )
3837a1i 10 . . . . . 6  |-  ( ph  ->  ( x  e.  {
x  e.  X  |  B  <  ( F `  x ) }  <->  ( x  e.  X  /\  B  < 
( F `  x
) ) ) )
3932, 36, 383bitr4d 276 . . . . 5  |-  ( ph  ->  ( x  e.  ( `' F " ( B (,)  +oo ) )  <->  x  e.  { x  e.  X  |  B  <  ( F `  x ) } ) )
401, 39alrimi 1745 . . . 4  |-  ( ph  ->  A. x ( x  e.  ( `' F " ( B (,)  +oo ) )  <->  x  e.  { x  e.  X  |  B  <  ( F `  x ) } ) )
41 rfcnpre1.2 . . . . . . 7  |-  F/_ x F
4241nfcnv 4860 . . . . . 6  |-  F/_ x `' F
43 rfcnpre1.1 . . . . . . 7  |-  F/_ x B
44 nfcv 2419 . . . . . . 7  |-  F/_ x (,)
45 nfcv 2419 . . . . . . 7  |-  F/_ x  +oo
4643, 44, 45nfov 5881 . . . . . 6  |-  F/_ x
( B (,)  +oo )
4742, 46nfima 5020 . . . . 5  |-  F/_ x
( `' F "
( B (,)  +oo ) )
48 nfrab1 2720 . . . . 5  |-  F/_ x { x  e.  X  |  B  <  ( F `
 x ) }
4947, 48cleqf 2443 . . . 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 ) } ) )
5040, 49sylibr 203 . . 3  |-  ( ph  ->  ( `' F "
( B (,)  +oo ) )  =  {
x  e.  X  |  B  <  ( F `  x ) } )
51 rfcnpre1.6 . . 3  |-  A  =  { x  e.  X  |  B  <  ( F `
 x ) }
5250, 51syl6eqr 2333 . 2  |-  ( ph  ->  ( `' F "
( B (,)  +oo ) )  =  A )
53 iooretop 18275 . . . . . 6  |-  ( B (,)  +oo )  e.  (
topGen `  ran  (,) )
5453, 12eleqtrri 2356 . . . . 5  |-  ( B (,)  +oo )  e.  K
5554a1i 10 . . . 4  |-  ( ph  ->  ( B (,)  +oo )  e.  K )
563, 55jca 518 . . 3  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  /\  ( B (,)  +oo )  e.  K ) )
57 cnima 16994 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( B (,)  +oo )  e.  K )  ->  ( `' F " ( B (,)  +oo ) )  e.  J )
5856, 57syl 15 . 2  |-  ( ph  ->  ( `' F "
( B (,)  +oo ) )  e.  J
)
5952, 58eqeltrrd 2358 1  |-  ( ph  ->  A  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406   A.wral 2543   {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    +oocpnf 8864   RR*cxr 8866    < clt 8867   (,)cioo 10656   topGenctg 13342   Topctop 16631  TopOnctopon 16632    Cn ccn 16954
This theorem is referenced by:  stoweidlem46  27795
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-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957
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