Theorem rfcnpre4 39193

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.

Step	Hyp	Ref	Expression
1		rfcnpre4.2	. . . . . . . 8 |- K = ( topGen ` ran (,) )
2		rfcnpre4.3	. . . . . . . 8 |- T = U. J
3		eqid 2622	. . . . . . . 8 |- ( J Cn K ) = ( J Cn K )
4		rfcnpre4.6	. . . . . . . 8 |- ( ph -> F e. ( J Cn K ) )
5	1, 2, 3, 4	fcnre 39184	. . . . . . 7 |- ( ph -> F : T --> RR )
6		ffn 6045	. . . . . . 7 |- ( F : T --> RR -> F Fn T )
7		elpreima 6337	. . . . . . 7 |- ( F Fn T -> ( s e. ( `' F " ( -oo (,] B ) ) <-> ( s e. T /\ ( F ` s ) e. ( -oo (,] B ) ) ) )
8	5, 6, 7	3syl 18	. . . . . 6 |- ( ph -> ( s e. ( `' F " ( -oo (,] B ) ) <-> ( s e. T /\ ( F ` s ) e. ( -oo (,] B ) ) ) )
9		mnfxr 10096	. . . . . . . . 9 |- -oo e. RR*
10		rfcnpre4.5	. . . . . . . . . . 11 |- ( ph -> B e. RR )
11	10	rexrd 10089	. . . . . . . . . 10 |- ( ph -> B e. RR* )
12	11	adantr 481	. . . . . . . . 9 |- ( ( ph /\ s e. T ) -> B e. RR* )
13		elioc1 12217	. . . . . . . . 9 |- ( ( -oo e. RR* /\ B e. RR* ) -> ( ( F ` s ) e. ( -oo (,] B ) <-> ( ( F ` s ) e. RR* /\ -oo < ( F ` s ) /\ ( F ` s ) <_ B ) ) )
14	9, 12, 13	sylancr 695	. . . . . . . 8 |- ( ( ph /\ s e. T ) -> ( ( F ` s ) e. ( -oo (,] B ) <-> ( ( F ` s ) e. RR* /\ -oo < ( F ` s ) /\ ( F ` s ) <_ B ) ) )
15		simpr3 1069	. . . . . . . . 9 |- ( ( ( ph /\ s e. T ) /\ ( ( F ` s ) e. RR* /\ -oo < ( F ` s ) /\ ( F ` s ) <_ B ) ) -> ( F ` s ) <_ B )
16	5	ffvelrnda 6359	. . . . . . . . . . . 12 |- ( ( ph /\ s e. T ) -> ( F ` s ) e. RR )
17	16	rexrd 10089	. . . . . . . . . . 11 |- ( ( ph /\ s e. T ) -> ( F ` s ) e. RR* )
18	17	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ s e. T ) /\ ( F ` s ) <_ B ) -> ( F ` s ) e. RR* )
19	16	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ s e. T ) /\ ( F ` s ) <_ B ) -> ( F ` s ) e. RR )
20		mnflt 11957	. . . . . . . . . . 11 |- ( ( F ` s ) e. RR -> -oo < ( F ` s ) )
21	19, 20	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ s e. T ) /\ ( F ` s ) <_ B ) -> -oo < ( F ` s ) )
22		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ s e. T ) /\ ( F ` s ) <_ B ) -> ( F ` s ) <_ B )
23	18, 21, 22	3jca 1242	. . . . . . . . 9 |- ( ( ( ph /\ s e. T ) /\ ( F ` s ) <_ B ) -> ( ( F ` s ) e. RR* /\ -oo < ( F ` s ) /\ ( F ` s ) <_ B ) )
24	15, 23	impbida 877	. . . . . . . 8 |- ( ( ph /\ s e. T ) -> ( ( ( F ` s ) e. RR* /\ -oo < ( F ` s ) /\ ( F ` s ) <_ B ) <-> ( F ` s ) <_ B ) )
25	14, 24	bitrd 268	. . . . . . 7 |- ( ( ph /\ s e. T ) -> ( ( F ` s ) e. ( -oo (,] B ) <-> ( F ` s ) <_ B ) )
26	25	pm5.32da 673	. . . . . 6 |- ( ph -> ( ( s e. T /\ ( F ` s ) e. ( -oo (,] B ) ) <-> ( s e. T /\ ( F ` s ) <_ B ) ) )
27	8, 26	bitrd 268	. . . . 5 |- ( ph -> ( s e. ( `' F " ( -oo (,] B ) ) <-> ( s e. T /\ ( F ` s ) <_ B ) ) )
28		nfcv 2764	. . . . . 6 |- F/_ t s
29		nfcv 2764	. . . . . 6 |- F/_ t T
30		rfcnpre4.1	. . . . . . . 8 |- F/_ t F
31	30, 28	nffv 6198	. . . . . . 7 |- F/_ t ( F ` s )
32		nfcv 2764	. . . . . . 7 |- F/_ t <_
33		nfcv 2764	. . . . . . 7 |- F/_ t B
34	31, 32, 33	nfbr 4699	. . . . . 6 |- F/ t ( F ` s ) <_ B
35		fveq2 6191	. . . . . . 7 |- ( t = s -> ( F ` t ) = ( F ` s ) )
36	35	breq1d 4663	. . . . . 6 |- ( t = s -> ( ( F ` t ) <_ B <-> ( F ` s ) <_ B ) )
37	28, 29, 34, 36	elrabf 3360	. . . . 5 |- ( s e. { t e. T | ( F ` t ) <_ B } <-> ( s e. T /\ ( F ` s ) <_ B ) )
38	27, 37	syl6bbr 278	. . . 4 |- ( ph -> ( s e. ( `' F " ( -oo (,] B ) ) <-> s e. { t e. T | ( F ` t ) <_ B } ) )
39	38	eqrdv 2620	. . 3 |- ( ph -> ( `' F " ( -oo (,] B ) ) = { t e. T | ( F ` t ) <_ B } )
40		rfcnpre4.4	. . 3 |- A = { t e. T | ( F ` t ) <_ B }
41	39, 40	syl6eqr 2674	. 2 |- ( ph -> ( `' F " ( -oo (,] B ) ) = A )
42		iocmnfcld 22572	. . . . 5 |- ( B e. RR -> ( -oo (,] B ) e. ( Clsd ` ( topGen ` ran (,) ) ) )
43	10, 42	syl 17	. . . 4 |- ( ph -> ( -oo (,] B ) e. ( Clsd ` ( topGen ` ran (,) ) ) )
44	1	fveq2i 6194	. . . 4 |- ( Clsd ` K ) = ( Clsd ` ( topGen ` ran (,) ) )
45	43, 44	syl6eleqr 2712	. . 3 |- ( ph -> ( -oo (,] B ) e. ( Clsd ` K ) )
46		cnclima 21072	. . 3 |- ( ( F e. ( J Cn K ) /\ ( -oo (,] B ) e. ( Clsd ` K ) ) -> ( `' F " ( -oo (,] B ) ) e. ( Clsd ` J ) )
47	4, 45, 46	syl2anc 693	. 2 |- ( ph -> ( `' F " ( -oo (,] B ) ) e. ( Clsd ` J ) )
48	41, 47	eqeltrrd 2702	1 |- ( ph -> A e. ( Clsd ` J ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   F/_wnfc 2751   {crab 2916   U.cuni 4436   class class class wbr 4653   `'ccnv 5113   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175   (,]cioc 12176   topGenctg 16098   Clsdccld 20820   Cn ccn 21028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-ioo 12179  df-ioc 12180  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-cn 21031

This theorem is referenced by:  stoweidlem59  40276