Theorem rfcnpre1 36995

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 5033	. . . . 5 |- F/_ x `' F
4		rfcnpre1.1	. . . . . 6 |- F/_ x B
5		nfcv 2591	. . . . . 6 |- F/_ x (,)
6		nfcv 2591	. . . . . 6 |- F/_ x +oo
7	4, 5, 6	nfov 6331	. . . . 5 |- F/_ x ( B (,) +oo )
8	3, 7	nfima 5196	. . . 4 |- F/_ x ( `' F " ( B (,) +oo ) )
9		nfrab1 3016	. . . 4 |- F/_ x { x e. X | B < ( F ` x ) }
10		rfcnpre1.8	. . . . . . . . . 10 |- ( ph -> F e. ( J Cn K ) )
11		cntop1 20191	. . . . . . . . . . . . . 14 |- ( F e. ( J Cn K ) -> J e. Top )
12	10, 11	syl 17	. . . . . . . . . . . . 13 |- ( ph -> J e. Top )
13		rfcnpre1.5	. . . . . . . . . . . . 13 |- X = U. J
14	12, 13	jctir 540	. . . . . . . . . . . 12 |- ( ph -> ( J e. Top /\ X = U. J ) )
15		istopon 19875	. . . . . . . . . . . 12 |- ( J e. (TopOn ` X ) <-> ( J e. Top /\ X = U. J ) )
16	14, 15	sylibr 215	. . . . . . . . . . 11 |- ( ph -> J e. (TopOn ` X ) )
17		rfcnpre1.4	. . . . . . . . . . . 12 |- K = ( topGen ` ran (,) )
18		retopon 21699	. . . . . . . . . . . 12 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
19	17, 18	eqeltri 2513	. . . . . . . . . . 11 |- K e. (TopOn ` RR )
20		iscn 20186	. . . . . . . . . . 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 ) ) )
21	16, 19, 20	sylancl 666	. . . . . . . . . 10 |- ( ph -> ( F e. ( J Cn K ) <-> ( F : X --> RR /\ A. y e. K ( `' F " y ) e. J ) ) )
22	10, 21	mpbid 213	. . . . . . . . 9 |- ( ph -> ( F : X --> RR /\ A. y e. K ( `' F " y ) e. J ) )
23	22	simpld 460	. . . . . . . 8 |- ( ph -> F : X --> RR )
24	23	fnvinran 36990	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. RR )
25		rfcnpre1.7	. . . . . . . . 9 |- ( ph -> B e. RR* )
26		elioopnf 11728	. . . . . . . . 9 |- ( B e. RR* -> ( ( F ` x ) e. ( B (,) +oo ) <-> ( ( F ` x ) e. RR /\ B < ( F ` x ) ) ) )
27	25, 26	syl 17	. . . . . . . 8 |- ( ph -> ( ( F ` x ) e. ( B (,) +oo ) <-> ( ( F ` x ) e. RR /\ B < ( F ` x ) ) ) )
28	27	baibd 917	. . . . . . 7 |- ( ( ph /\ ( F ` x ) e. RR ) -> ( ( F ` x ) e. ( B (,) +oo ) <-> B < ( F ` x ) ) )
29	24, 28	syldan 472	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( F ` x ) e. ( B (,) +oo ) <-> B < ( F ` x ) ) )
30	29	pm5.32da 645	. . . . 5 |- ( ph -> ( ( x e. X /\ ( F ` x ) e. ( B (,) +oo ) ) <-> ( x e. X /\ B < ( F ` x ) ) ) )
31		ffn 5746	. . . . . 6 |- ( F : X --> RR -> F Fn X )
32		elpreima 6017	. . . . . 6 |- ( F Fn X -> ( x e. ( `' F " ( B (,) +oo ) ) <-> ( x e. X /\ ( F ` x ) e. ( B (,) +oo ) ) ) )
33	23, 31, 32	3syl 18	. . . . 5 |- ( ph -> ( x e. ( `' F " ( B (,) +oo ) ) <-> ( x e. X /\ ( F ` x ) e. ( B (,) +oo ) ) ) )
34		rabid 3012	. . . . . 6 |- ( x e. { x e. X | B < ( F ` x ) } <-> ( x e. X /\ B < ( F ` x ) ) )
35	34	a1i 11	. . . . 5 |- ( ph -> ( x e. { x e. X | B < ( F ` x ) } <-> ( x e. X /\ B < ( F ` x ) ) ) )
36	30, 33, 35	3bitr4d 288	. . . 4 |- ( ph -> ( x e. ( `' F " ( B (,) +oo ) ) <-> x e. { x e. X | B < ( F ` x ) } ) )
37	1, 8, 9, 36	eqrd 3488	. . 3 |- ( ph -> ( `' F " ( B (,) +oo ) ) = { x e. X | B < ( F ` x ) } )
38		rfcnpre1.6	. . 3 |- A = { x e. X | B < ( F ` x ) }
39	37, 38	syl6eqr 2488	. 2 |- ( ph -> ( `' F " ( B (,) +oo ) ) = A )
40		iooretop 21701	. . . 4 |- ( B (,) +oo ) e. ( topGen ` ran (,) )
41	40, 17	eleqtrri 2516	. . 3 |- ( B (,) +oo ) e. K
42		cnima 20216	. . 3 |- ( ( F e. ( J Cn K ) /\ ( B (,) +oo ) e. K ) -> ( `' F " ( B (,) +oo ) ) e. J )
43	10, 41, 42	sylancl 666	. 2 |- ( ph -> ( `' F " ( B (,) +oo ) ) e. J )
44	39, 43	eqeltrrd 2518	1 |- ( ph -> A e. J )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   F/wnf 1663   e. wcel 1870   F/_wnfc 2577   A.wral 2782   {crab 2786   U.cuni 4222   class class class wbr 4426   `'ccnv 4853   ran crn 4855   "cima 4857   Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   RRcr 9537   +oocpnf 9671   RR*cxr 9673   < clt 9674   (,)cioo 11635   topGenctg 15299   Topctop 19852  TopOnctopon 19853   Cn ccn 20175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-ioo 11639  df-topgen 15305  df-top 19856  df-bases 19857  df-topon 19858  df-cn 20178

This theorem is referenced by:  stoweidlem46  37491