Theorem rfcnpre1 39178

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	. . . 4 |- F/ x ph
2		rfcnpre1.2	. . . . . 6 |- F/_ x F
3	2	nfcnv 5301	. . . . 5 |- F/_ x `' F
4		rfcnpre1.1	. . . . . 6 |- F/_ x B
5		nfcv 2764	. . . . . 6 |- F/_ x (,)
6		nfcv 2764	. . . . . 6 |- F/_ x +oo
7	4, 5, 6	nfov 6676	. . . . 5 |- F/_ x ( B (,) +oo )
8	3, 7	nfima 5474	. . . 4 |- F/_ x ( `' F " ( B (,) +oo ) )
9		nfrab1 3122	. . . 4 |- F/_ x { x e. X | B < ( F ` x ) }
10		rfcnpre1.8	. . . . . . . . . 10 |- ( ph -> F e. ( J Cn K ) )
11		cntop1 21044	. . . . . . . . . . . . 13 |- ( F e. ( J Cn K ) -> J e. Top )
12	10, 11	syl 17	. . . . . . . . . . . 12 |- ( ph -> J e. Top )
13		rfcnpre1.5	. . . . . . . . . . . 12 |- X = U. J
14		istopon 20717	. . . . . . . . . . . 12 |- ( J e. (TopOn ` X ) <-> ( J e. Top /\ X = U. J ) )
15	12, 13, 14	sylanblrc 697	. . . . . . . . . . 11 |- ( ph -> J e. (TopOn ` X ) )
16		rfcnpre1.4	. . . . . . . . . . . 12 |- K = ( topGen ` ran (,) )
17		retopon 22567	. . . . . . . . . . . 12 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
18	16, 17	eqeltri 2697	. . . . . . . . . . 11 |- K e. (TopOn ` RR )
19		iscn 21039	. . . . . . . . . . 11 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` RR ) ) -> ( F e. ( J Cn K ) <-> ( F : X --> RR /\ A. y e. K ( `' F " y ) e. J ) ) )
20	15, 18, 19	sylancl 694	. . . . . . . . . 10 |- ( ph -> ( F e. ( J Cn K ) <-> ( F : X --> RR /\ A. y e. K ( `' F " y ) e. J ) ) )
21	10, 20	mpbid 222	. . . . . . . . 9 |- ( ph -> ( F : X --> RR /\ A. y e. K ( `' F " y ) e. J ) )
22	21	simpld 475	. . . . . . . 8 |- ( ph -> F : X --> RR )
23	22	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. RR )
24		rfcnpre1.7	. . . . . . . . 9 |- ( ph -> B e. RR* )
25		elioopnf 12267	. . . . . . . . 9 |- ( B e. RR* -> ( ( F ` x ) e. ( B (,) +oo ) <-> ( ( F ` x ) e. RR /\ B < ( F ` x ) ) ) )
26	24, 25	syl 17	. . . . . . . 8 |- ( ph -> ( ( F ` x ) e. ( B (,) +oo ) <-> ( ( F ` x ) e. RR /\ B < ( F ` x ) ) ) )
27	26	baibd 948	. . . . . . 7 |- ( ( ph /\ ( F ` x ) e. RR ) -> ( ( F ` x ) e. ( B (,) +oo ) <-> B < ( F ` x ) ) )
28	23, 27	syldan 487	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( F ` x ) e. ( B (,) +oo ) <-> B < ( F ` x ) ) )
29	28	pm5.32da 673	. . . . 5 |- ( ph -> ( ( x e. X /\ ( F ` x ) e. ( B (,) +oo ) ) <-> ( x e. X /\ B < ( F ` x ) ) ) )
30		ffn 6045	. . . . . 6 |- ( F : X --> RR -> F Fn X )
31		elpreima 6337	. . . . . 6 |- ( F Fn X -> ( x e. ( `' F " ( B (,) +oo ) ) <-> ( x e. X /\ ( F ` x ) e. ( B (,) +oo ) ) ) )
32	22, 30, 31	3syl 18	. . . . 5 |- ( ph -> ( x e. ( `' F " ( B (,) +oo ) ) <-> ( x e. X /\ ( F ` x ) e. ( B (,) +oo ) ) ) )
33		rabid 3116	. . . . . 6 |- ( x e. { x e. X | B < ( F ` x ) } <-> ( x e. X /\ B < ( F ` x ) ) )
34	33	a1i 11	. . . . 5 |- ( ph -> ( x e. { x e. X | B < ( F ` x ) } <-> ( x e. X /\ B < ( F ` x ) ) ) )
35	29, 32, 34	3bitr4d 300	. . . 4 |- ( ph -> ( x e. ( `' F " ( B (,) +oo ) ) <-> x e. { x e. X | B < ( F ` x ) } ) )
36	1, 8, 9, 35	eqrd 3622	. . 3 |- ( ph -> ( `' F " ( B (,) +oo ) ) = { x e. X | B < ( F ` x ) } )
37		rfcnpre1.6	. . 3 |- A = { x e. X | B < ( F ` x ) }
38	36, 37	syl6eqr 2674	. 2 |- ( ph -> ( `' F " ( B (,) +oo ) ) = A )
39		iooretop 22569	. . . 4 |- ( B (,) +oo ) e. ( topGen ` ran (,) )
40	39, 16	eleqtrri 2700	. . 3 |- ( B (,) +oo ) e. K
41		cnima 21069	. . 3 |- ( ( F e. ( J Cn K ) /\ ( B (,) +oo ) e. K ) -> ( `' F " ( B (,) +oo ) ) e. J )
42	10, 40, 41	sylancl 694	. 2 |- ( ph -> ( `' F " ( B (,) +oo ) ) e. J )
43	38, 42	eqeltrrd 2702	1 |- ( ph -> A e. J )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   A.wral 2912   {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   +oocpnf 10071   RR*cxr 10073   < clt 10074   (,)cioo 12175   topGenctg 16098   Topctop 20698  TopOnctopon 20715   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-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031

This theorem is referenced by:  stoweidlem46  40263