Theorem rfcnpre1 37340

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 5013	. . . . 5 |- F/_ x `' F
4		rfcnpre1.1	. . . . . 6 |- F/_ x B
5		nfcv 2592	. . . . . 6 |- F/_ x (,)
6		nfcv 2592	. . . . . 6 |- F/_ x +oo
7	4, 5, 6	nfov 6316	. . . . 5 |- F/_ x ( B (,) +oo )
8	3, 7	nfima 5176	. . . 4 |- F/_ x ( `' F " ( B (,) +oo ) )
9		nfrab1 2971	. . . 4 |- F/_ x { x e. X | B < ( F ` x ) }
10		rfcnpre1.8	. . . . . . . . . 10 |- ( ph -> F e. ( J Cn K ) )
11		cntop1 20256	. . . . . . . . . . . . . 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 541	. . . . . . . . . . . 12 |- ( ph -> ( J e. Top /\ X = U. J ) )
15		istopon 19940	. . . . . . . . . . . 12 |- ( J e. (TopOn ` X ) <-> ( J e. Top /\ X = U. J ) )
16	14, 15	sylibr 216	. . . . . . . . . . 11 |- ( ph -> J e. (TopOn ` X ) )
17		rfcnpre1.4	. . . . . . . . . . . 12 |- K = ( topGen ` ran (,) )
18		retopon 21784	. . . . . . . . . . . 12 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
19	17, 18	eqeltri 2525	. . . . . . . . . . 11 |- K e. (TopOn ` RR )
20		iscn 20251	. . . . . . . . . . 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 668	. . . . . . . . . 10 |- ( ph -> ( F e. ( J Cn K ) <-> ( F : X --> RR /\ A. y e. K ( `' F " y ) e. J ) ) )
22	10, 21	mpbid 214	. . . . . . . . 9 |- ( ph -> ( F : X --> RR /\ A. y e. K ( `' F " y ) e. J ) )
23	22	simpld 461	. . . . . . . 8 |- ( ph -> F : X --> RR )
24	23	fnvinran 37335	. . . . . . 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 920	. . . . . . 7 |- ( ( ph /\ ( F ` x ) e. RR ) -> ( ( F ` x ) e. ( B (,) +oo ) <-> B < ( F ` x ) ) )
29	24, 28	syldan 473	. . . . . 6 |- ( ( ph /\ x e. X ) -> ( ( F ` x ) e. ( B (,) +oo ) <-> B < ( F ` x ) ) )
30	29	pm5.32da 647	. . . . 5 |- ( ph -> ( ( x e. X /\ ( F ` x ) e. ( B (,) +oo ) ) <-> ( x e. X /\ B < ( F ` x ) ) ) )
31		ffn 5728	. . . . . 6 |- ( F : X --> RR -> F Fn X )
32		elpreima 6002	. . . . . 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 2967	. . . . . 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 289	. . . 4 |- ( ph -> ( x e. ( `' F " ( B (,) +oo ) ) <-> x e. { x e. X | B < ( F ` x ) } ) )
37	1, 8, 9, 36	eqrd 3450	. . 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 2503	. 2 |- ( ph -> ( `' F " ( B (,) +oo ) ) = A )
40		iooretop 21786	. . . 4 |- ( B (,) +oo ) e. ( topGen ` ran (,) )
41	40, 17	eleqtrri 2528	. . 3 |- ( B (,) +oo ) e. K
42		cnima 20281	. . 3 |- ( ( F e. ( J Cn K ) /\ ( B (,) +oo ) e. K ) -> ( `' F " ( B (,) +oo ) ) e. J )
43	10, 41, 42	sylancl 668	. 2 |- ( ph -> ( `' F " ( B (,) +oo ) ) e. J )
44	39, 43	eqeltrrd 2530	1 |- ( ph -> A e. J )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   F/wnf 1667   e. wcel 1887   F/_wnfc 2579   A.wral 2737   {crab 2741   U.cuni 4198   class class class wbr 4402   `'ccnv 4833   ran crn 4835   "cima 4837   Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   RRcr 9538   +oocpnf 9672   RR*cxr 9674   < clt 9675   (,)cioo 11635   topGenctg 15336   Topctop 19917  TopOnctopon 19918   Cn ccn 20240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-ioo 11639  df-topgen 15342  df-top 19921  df-bases 19922  df-topon 19923  df-cn 20243

This theorem is referenced by:  stoweidlem46  37907