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Theorem rfcnpre4 31649
Description: If F is a continuous function with respect to the standard topology, then the preimage A of the values smaller or equal than a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
rfcnpre4.1  |-  F/_ t F
rfcnpre4.2  |-  K  =  ( topGen `  ran  (,) )
rfcnpre4.3  |-  T  = 
U. J
rfcnpre4.4  |-  A  =  { t  e.  T  |  ( F `  t )  <_  B }
rfcnpre4.5  |-  ( ph  ->  B  e.  RR )
rfcnpre4.6  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
Assertion
Ref Expression
rfcnpre4  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
Distinct variable groups:    t, B    t, T
Allowed substitution hints:    ph( t)    A( t)    F( t)    J( t)    K( t)

Proof of Theorem rfcnpre4
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 rfcnpre4.2 . . . . . . . 8  |-  K  =  ( topGen `  ran  (,) )
2 rfcnpre4.3 . . . . . . . 8  |-  T  = 
U. J
3 eqid 2454 . . . . . . . 8  |-  ( J  Cn  K )  =  ( J  Cn  K
)
4 rfcnpre4.6 . . . . . . . 8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
51, 2, 3, 4fcnre 31640 . . . . . . 7  |-  ( ph  ->  F : T --> RR )
6 ffn 5713 . . . . . . 7  |-  ( F : T --> RR  ->  F  Fn  T )
7 elpreima 5983 . . . . . . 7  |-  ( F  Fn  T  ->  (
s  e.  ( `' F " ( -oo (,] B ) )  <->  ( s  e.  T  /\  ( F `  s )  e.  ( -oo (,] B
) ) ) )
85, 6, 73syl 20 . . . . . 6  |-  ( ph  ->  ( s  e.  ( `' F " ( -oo (,] B ) )  <->  ( s  e.  T  /\  ( F `  s )  e.  ( -oo (,] B
) ) ) )
9 mnfxr 11326 . . . . . . . . 9  |- -oo  e.  RR*
10 rfcnpre4.5 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR )
1110rexrd 9632 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR* )
1211adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  T )  ->  B  e.  RR* )
13 elioc1 11574 . . . . . . . . 9  |-  ( ( -oo  e.  RR*  /\  B  e.  RR* )  ->  (
( F `  s
)  e.  ( -oo (,] B )  <->  ( ( F `  s )  e.  RR*  /\ -oo  <  ( F `  s )  /\  ( F `  s )  <_  B
) ) )
149, 12, 13sylancr 661 . . . . . . . 8  |-  ( (
ph  /\  s  e.  T )  ->  (
( F `  s
)  e.  ( -oo (,] B )  <->  ( ( F `  s )  e.  RR*  /\ -oo  <  ( F `  s )  /\  ( F `  s )  <_  B
) ) )
15 simpr3 1002 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  T )  /\  (
( F `  s
)  e.  RR*  /\ -oo  <  ( F `  s
)  /\  ( F `  s )  <_  B
) )  ->  ( F `  s )  <_  B )
165fnvinran 31629 . . . . . . . . . . . 12  |-  ( (
ph  /\  s  e.  T )  ->  ( F `  s )  e.  RR )
1716rexrd 9632 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  T )  ->  ( F `  s )  e.  RR* )
1817adantr 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  ( F `  s )  e.  RR* )
1916adantr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  ( F `  s )  e.  RR )
20 mnflt 11336 . . . . . . . . . . 11  |-  ( ( F `  s )  e.  RR  -> -oo  <  ( F `  s ) )
2119, 20syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  -> -oo  <  ( F `  s ) )
22 simpr 459 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  ( F `  s )  <_  B )
2318, 21, 223jca 1174 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  T )  /\  ( F `  s )  <_  B )  ->  (
( F `  s
)  e.  RR*  /\ -oo  <  ( F `  s
)  /\  ( F `  s )  <_  B
) )
2415, 23impbida 830 . . . . . . . 8  |-  ( (
ph  /\  s  e.  T )  ->  (
( ( F `  s )  e.  RR*  /\ -oo  <  ( F `  s )  /\  ( F `  s )  <_  B )  <->  ( F `  s )  <_  B
) )
2514, 24bitrd 253 . . . . . . 7  |-  ( (
ph  /\  s  e.  T )  ->  (
( F `  s
)  e.  ( -oo (,] B )  <->  ( F `  s )  <_  B
) )
2625pm5.32da 639 . . . . . 6  |-  ( ph  ->  ( ( s  e.  T  /\  ( F `
 s )  e.  ( -oo (,] B
) )  <->  ( s  e.  T  /\  ( F `  s )  <_  B ) ) )
278, 26bitrd 253 . . . . 5  |-  ( ph  ->  ( s  e.  ( `' F " ( -oo (,] B ) )  <->  ( s  e.  T  /\  ( F `  s )  <_  B ) ) )
28 nfcv 2616 . . . . . 6  |-  F/_ t
s
29 nfcv 2616 . . . . . 6  |-  F/_ t T
30 rfcnpre4.1 . . . . . . . 8  |-  F/_ t F
3130, 28nffv 5855 . . . . . . 7  |-  F/_ t
( F `  s
)
32 nfcv 2616 . . . . . . 7  |-  F/_ t  <_
33 nfcv 2616 . . . . . . 7  |-  F/_ t B
3431, 32, 33nfbr 4483 . . . . . 6  |-  F/ t ( F `  s
)  <_  B
35 fveq2 5848 . . . . . . 7  |-  ( t  =  s  ->  ( F `  t )  =  ( F `  s ) )
3635breq1d 4449 . . . . . 6  |-  ( t  =  s  ->  (
( F `  t
)  <_  B  <->  ( F `  s )  <_  B
) )
3728, 29, 34, 36elrabf 3252 . . . . 5  |-  ( s  e.  { t  e.  T  |  ( F `
 t )  <_  B }  <->  ( s  e.  T  /\  ( F `
 s )  <_  B ) )
3827, 37syl6bbr 263 . . . 4  |-  ( ph  ->  ( s  e.  ( `' F " ( -oo (,] B ) )  <->  s  e.  { t  e.  T  | 
( F `  t
)  <_  B }
) )
3938eqrdv 2451 . . 3  |-  ( ph  ->  ( `' F "
( -oo (,] B ) )  =  { t  e.  T  |  ( F `  t )  <_  B } )
40 rfcnpre4.4 . . 3  |-  A  =  { t  e.  T  |  ( F `  t )  <_  B }
4139, 40syl6eqr 2513 . 2  |-  ( ph  ->  ( `' F "
( -oo (,] B ) )  =  A )
42 iocmnfcld 21442 . . . . 5  |-  ( B  e.  RR  ->  ( -oo (,] B )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )
4310, 42syl 16 . . . 4  |-  ( ph  ->  ( -oo (,] B
)  e.  ( Clsd `  ( topGen `  ran  (,) )
) )
441fveq2i 5851 . . . 4  |-  ( Clsd `  K )  =  (
Clsd `  ( topGen ` 
ran  (,) ) )
4543, 44syl6eleqr 2553 . . 3  |-  ( ph  ->  ( -oo (,] B
)  e.  ( Clsd `  K ) )
46 cnclima 19936 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( -oo (,] B )  e.  ( Clsd `  K
) )  ->  ( `' F " ( -oo (,] B ) )  e.  ( Clsd `  J
) )
474, 45, 46syl2anc 659 . 2  |-  ( ph  ->  ( `' F "
( -oo (,] B ) )  e.  ( Clsd `  J ) )
4841, 47eqeltrrd 2543 1  |-  ( ph  ->  A  e.  ( Clsd `  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   F/_wnfc 2602   {crab 2808   U.cuni 4235   class class class wbr 4439   `'ccnv 4987   ran crn 4989   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   RRcr 9480   -oocmnf 9615   RR*cxr 9616    < clt 9617    <_ cle 9618   (,)cioo 11532   (,]cioc 11533   topGenctg 14927   Clsdccld 19684    Cn ccn 19892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-ioo 11536  df-ioc 11537  df-topgen 14933  df-top 19566  df-bases 19568  df-topon 19569  df-cld 19687  df-cn 19895
This theorem is referenced by:  stoweidlem59  32080
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