Theorem rfcnpre1 29741

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.

Step	Hyp	Ref	Expression
1		rfcnpre1.3	. . . . 5 |- F/ x ph
2		rfcnpre1.8	. . . . . . . . . . 11 |- ( ph -> F e. ( J Cn K ) )
3		cntop1 18844	. . . . . . . . . . . . . . 15 |- ( F e. ( J Cn K ) -> J e. Top )
4	2, 3	syl 16	. . . . . . . . . . . . . 14 |- ( ph -> J e. Top )
5		rfcnpre1.5	. . . . . . . . . . . . . 14 |- X = U. J
6	4, 5	jctir 538	. . . . . . . . . . . . 13 |- ( ph -> ( J e. Top /\ X = U. J ) )
7		istopon 18530	. . . . . . . . . . . . 13 |- ( J e. (TopOn ` X ) <-> ( J e. Top /\ X = U. J ) )
8	6, 7	sylibr 212	. . . . . . . . . . . 12 |- ( ph -> J e. (TopOn ` X ) )
9		rfcnpre1.4	. . . . . . . . . . . . 13 |- K = ( topGen ` ran (,) )
10		retopon 20342	. . . . . . . . . . . . 13 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
11	9, 10	eqeltri 2513	. . . . . . . . . . . 12 |- K e. (TopOn ` RR )
12		iscn 18839	. . . . . . . . . . . 12 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> RR /\ A. y e. K ( `' F " y ) e. J ) ) )
13	8, 11, 12	sylancl 662	. . . . . . . . . . 11 |- ( ph -> ( F e. ( J Cn K ) <-> ( F : X --> RR /\ A. y e. K ( `' F " y ) e. J ) ) )
14	2, 13	mpbid 210	. . . . . . . . . 10 |- ( ph -> ( F : X --> RR /\ A. y e. K ( `' F " y ) e. J ) )
15	14	simpld 459	. . . . . . . . 9 |- ( ph -> F : X --> RR )
16	15	fnvinran 29736	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. RR )
17		rfcnpre1.7	. . . . . . . . . 10 |- ( ph -> B e. RR* )
18		elioopnf 11383	. . . . . . . . . 10 |- ( B e. RR* -> ( ( F ` x ) e. ( B (,) +oo ) <-> ( ( F ` x ) e. RR /\ B < ( F ` x ) ) ) )
19	17, 18	syl 16	. . . . . . . . 9 |- ( ph -> ( ( F ` x ) e. ( B (,) +oo ) <-> ( ( F ` x ) e. RR /\ B < ( F ` x ) ) ) )
20	19	baibd 900	. . . . . . . 8 |- ( ( ph /\ ( F ` x ) e. RR ) -> ( ( F ` x ) e. ( B (,) +oo ) <-> B < ( F ` x ) ) )
21	16, 20	syldan 470	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( ( F ` x ) e. ( B (,) +oo ) <-> B < ( F ` x ) ) )
22	21	pm5.32da 641	. . . . . 6 |- ( ph -> ( ( x e. X /\ ( F ` x ) e. ( B (,) +oo ) ) <-> ( x e. X /\ B < ( F ` x ) ) ) )
23		ffn 5559	. . . . . . 7 |- ( F : X --> RR -> F Fn X )
24		elpreima 5823	. . . . . . 7 |- ( F Fn X -> ( x e. ( `' F " ( B (,) +oo ) ) <-> ( x e. X /\ ( F ` x ) e. ( B (,) +oo ) ) ) )
25	15, 23, 24	3syl 20	. . . . . 6 |- ( ph -> ( x e. ( `' F " ( B (,) +oo ) ) <-> ( x e. X /\ ( F ` x ) e. ( B (,) +oo ) ) ) )
26		rabid 2897	. . . . . . 7 |- ( x e. { x e. X | B < ( F ` x ) } <-> ( x e. X /\ B < ( F ` x ) ) )
27	26	a1i 11	. . . . . 6 |- ( ph -> ( x e. { x e. X | B < ( F ` x ) } <-> ( x e. X /\ B < ( F ` x ) ) ) )
28	22, 25, 27	3bitr4d 285	. . . . 5 |- ( ph -> ( x e. ( `' F " ( B (,) +oo ) ) <-> x e. { x e. X | B < ( F ` x ) } ) )
29	1, 28	alrimi 1811	. . . 4 |- ( ph -> A. x ( x e. ( `' F " ( B (,) +oo ) ) <-> x e. { x e. X | B < ( F ` x ) } ) )
30		rfcnpre1.2	. . . . . . 7 |- F/_ x F
31	30	nfcnv 5018	. . . . . 6 |- F/_ x `' F
32		rfcnpre1.1	. . . . . . 7 |- F/_ x B
33		nfcv 2579	. . . . . . 7 |- F/_ x (,)
34		nfcv 2579	. . . . . . 7 |- F/_ x +oo
35	32, 33, 34	nfov 6114	. . . . . 6 |- F/_ x ( B (,) +oo )
36	31, 35	nfima 5177	. . . . 5 |- F/_ x ( `' F " ( B (,) +oo ) )
37		nfrab1 2901	. . . . 5 |- F/_ x { x e. X | B < ( F ` x ) }
38	36, 37	cleqf 2603	. . . 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 ) } ) )
39	29, 38	sylibr 212	. . 3 |- ( ph -> ( `' F " ( B (,) +oo ) ) = { x e. X | B < ( F ` x ) } )
40		rfcnpre1.6	. . 3 |- A = { x e. X | B < ( F ` x ) }
41	39, 40	syl6eqr 2493	. 2 |- ( ph -> ( `' F " ( B (,) +oo ) ) = A )
42		iooretop 20345	. . . 4 |- ( B (,) +oo ) e. ( topGen ` ran (,) )
43	42, 9	eleqtrri 2516	. . 3 |- ( B (,) +oo ) e. K
44		cnima 18869	. . 3 |- ( ( F e. ( J Cn K ) /\ ( B (,) +oo ) e. K ) -> ( `' F " ( B (,) +oo ) ) e. J )
45	2, 43, 44	sylancl 662	. 2 |- ( ph -> ( `' F " ( B (,) +oo ) ) e. J )
46	41, 45	eqeltrrd 2518	1 |- ( ph -> A e. J )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1367   = wceq 1369   F/wnf 1589   e. wcel 1756   F/_wnfc 2566   A.wral 2715   {crab 2719   U.cuni 4091   class class class wbr 4292   `'ccnv 4839   ran crn 4841   "cima 4843   Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091   RRcr 9281   +oocpnf 9415   RR*cxr 9417   < clt 9418   (,)cioo 11300   topGenctg 14376   Topctop 18498  TopOnctopon 18499   Cn ccn 18828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-ioo 11304  df-topgen 14382  df-top 18503  df-bases 18505  df-topon 18506  df-cn 18831

This theorem is referenced by:  stoweidlem46  29841