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Theorem rfcnpre1 27013
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 17076 . . . . . . . . . . . . . . . . . . 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 16769 . . . . . . . . . . . . . . . . 17  |-  ( J  e.  (TopOn `  X
)  <->  ( J  e. 
Top  /\  X  =  U. J ) )
108, 9sylibr 203 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  J  e.  (TopOn `  X ) )
11 retopon 18374 . . . . . . . . . . . . . . . . . 18  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
12 rfcnpre1.4 . . . . . . . . . . . . . . . . . . 19  |-  K  =  ( topGen `  ran  (,) )
1312eleq1i 2421 . . . . . . . . . . . . . . . . . 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 17071 . . . . . . . . . . . . . . 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 5746 . . . . . . . . . 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 10829 . . . . . . . . . 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 5472 . . . . . . . 8  |-  ( F : X --> RR  ->  F  Fn  X )
3420, 33syl 15 . . . . . . 7  |-  ( ph  ->  F  Fn  X )
35 elpreima 5728 . . . . . . 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 2792 . . . . . . 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 1766 . . . 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 4942 . . . . . 6  |-  F/_ x `' F
43 rfcnpre1.1 . . . . . . 7  |-  F/_ x B
44 nfcv 2494 . . . . . . 7  |-  F/_ x (,)
45 nfcv 2494 . . . . . . 7  |-  F/_ x  +oo
4643, 44, 45nfov 5968 . . . . . 6  |-  F/_ x
( B (,)  +oo )
4742, 46nfima 5102 . . . . 5  |-  F/_ x
( `' F "
( B (,)  +oo ) )
48 nfrab1 2796 . . . . 5  |-  F/_ x { x  e.  X  |  B  <  ( F `
 x ) }
4947, 48cleqf 2518 . . . 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 2408 . 2  |-  ( ph  ->  ( `' F "
( B (,)  +oo ) )  =  A )
53 iooretop 18377 . . . . . 6  |-  ( B (,)  +oo )  e.  (
topGen `  ran  (,) )
5453, 12eleqtrri 2431 . . . . 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 17100 . . 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 2433 1  |-  ( ph  ->  A  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1540   F/wnf 1544    = wceq 1642    e. wcel 1710   F/_wnfc 2481   A.wral 2619   {crab 2623   U.cuni 3908   class class class wbr 4104   `'ccnv 4770   ran crn 4772   "cima 4774    Fn wfn 5332   -->wf 5333   ` cfv 5337  (class class class)co 5945   RRcr 8826    +oocpnf 8954   RR*cxr 8956    < clt 8957   (,)cioo 10748   topGenctg 13441   Topctop 16737  TopOnctopon 16738    Cn ccn 17060
This theorem is referenced by:  stoweidlem46  27118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-sup 7284  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-n0 10058  df-z 10117  df-uz 10323  df-q 10409  df-ioo 10752  df-topgen 13443  df-top 16742  df-bases 16744  df-topon 16745  df-cn 17063
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