Theorem rfcnpre1 30972

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 19504	. . . . . . . . . . . . . . 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 19190	. . . . . . . . . . . . 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 21002	. . . . . . . . . . . . 13 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
11	9, 10	eqeltri 2551	. . . . . . . . . . . 12 |- K e. (TopOn ` RR )
12		iscn 19499	. . . . . . . . . . . 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 30967	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. RR )
17		rfcnpre1.7	. . . . . . . . . 10 |- ( ph -> B e. RR* )
18		elioopnf 11614	. . . . . . . . . 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 907	. . . . . . . 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 5729	. . . . . . 7 |- ( F : X --> RR -> F Fn X )
24		elpreima 5999	. . . . . . 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 3038	. . . . . . 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 1825	. . . 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 5179	. . . . . 6 |- F/_ x `' F
32		rfcnpre1.1	. . . . . . 7 |- F/_ x B
33		nfcv 2629	. . . . . . 7 |- F/_ x (,)
34		nfcv 2629	. . . . . . 7 |- F/_ x +oo
35	32, 33, 34	nfov 6305	. . . . . 6 |- F/_ x ( B (,) +oo )
36	31, 35	nfima 5343	. . . . 5 |- F/_ x ( `' F " ( B (,) +oo ) )
37		nfrab1 3042	. . . . 5 |- F/_ x { x e. X | B < ( F ` x ) }
38	36, 37	cleqf 2656	. . . 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 2526	. 2 |- ( ph -> ( `' F " ( B (,) +oo ) ) = A )
42		iooretop 21005	. . . 4 |- ( B (,) +oo ) e. ( topGen ` ran (,) )
43	42, 9	eleqtrri 2554	. . 3 |- ( B (,) +oo ) e. K
44		cnima 19529	. . 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 2556	1 |- ( ph -> A e. J )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1377   = wceq 1379   F/wnf 1599   e. wcel 1767   F/_wnfc 2615   A.wral 2814   {crab 2818   U.cuni 4245   class class class wbr 4447   `'ccnv 4998   ran crn 5000   "cima 5002   Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   RRcr 9487   +oocpnf 9621   RR*cxr 9623   < clt 9624   (,)cioo 11525   topGenctg 14686   Topctop 19158  TopOnctopon 19159   Cn ccn 19488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-ioo 11529  df-topgen 14692  df-top 19163  df-bases 19165  df-topon 19166  df-cn 19491

This theorem is referenced by:  stoweidlem46  31346