Theorem cnpfcfi 23978

Description: Lemma for cnpfcf 23979. If a function is continuous at a point, it respects clustering there. (Contributed by Jeff Hankins, 20-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Assertion

Ref	Expression
cnpfcfi	⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝐿)‘𝐹))

Proof of Theorem cnpfcfi

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1137	. . 3 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fClus 𝐿))
2		eqid 2735	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	fclsfil 23948	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fClus 𝐿) → 𝐿 ∈ (Fil‘∪ 𝐽))
4	3	3ad2ant2 1134	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐿 ∈ (Fil‘∪ 𝐽))
5		fclsfnflim 23965	. . . 4 ⊢ (𝐿 ∈ (Fil‘∪ 𝐽) → (𝐴 ∈ (𝐽 fClus 𝐿) ↔ ∃𝑓 ∈ (Fil‘∪ 𝐽)(𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓))))
6	4, 5	syl 17	. . 3 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐴 ∈ (𝐽 fClus 𝐿) ↔ ∃𝑓 ∈ (Fil‘∪ 𝐽)(𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓))))
7	1, 6	mpbid 232	. 2 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∃𝑓 ∈ (Fil‘∪ 𝐽)(𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))
8		simpl1 1192	. . . . . 6 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐾 ∈ Top)
9		toptopon2 22856	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
10	8, 9	sylib 218	. . . . 5 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐾 ∈ (TopOn‘∪ 𝐾))
11		simprl 770	. . . . 5 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (Fil‘∪ 𝐽))
12		eqid 2735	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
13	2, 12	cnpf 23185	. . . . . . 7 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹:∪ 𝐽⟶∪ 𝐾)
14	13	3ad2ant3 1135	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
15	14	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐹:∪ 𝐽⟶∪ 𝐾)
16		flfssfcf 23976	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑓 ∈ (Fil‘∪ 𝐽) ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → ((𝐾 fLimf 𝑓)‘𝐹) ⊆ ((𝐾 fClusf 𝑓)‘𝐹))
17	10, 11, 15, 16	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → ((𝐾 fLimf 𝑓)‘𝐹) ⊆ ((𝐾 fClusf 𝑓)‘𝐹))
18	12	topopn 22844	. . . . . . . 8 ⊢ (𝐾 ∈ Top → ∪ 𝐾 ∈ 𝐾)
19	8, 18	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → ∪ 𝐾 ∈ 𝐾)
20	4	adantr 480	. . . . . . . 8 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐿 ∈ (Fil‘∪ 𝐽))
21		filfbas 23786	. . . . . . . 8 ⊢ (𝐿 ∈ (Fil‘∪ 𝐽) → 𝐿 ∈ (fBas‘∪ 𝐽))
22	20, 21	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐿 ∈ (fBas‘∪ 𝐽))
23		fmfil 23882	. . . . . . 7 ⊢ ((∪ 𝐾 ∈ 𝐾 ∧ 𝐿 ∈ (fBas‘∪ 𝐽) ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → ((∪ 𝐾 FilMap 𝐹)‘𝐿) ∈ (Fil‘∪ 𝐾))
24	19, 22, 15, 23	syl3anc 1373	. . . . . 6 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → ((∪ 𝐾 FilMap 𝐹)‘𝐿) ∈ (Fil‘∪ 𝐾))
25		filfbas 23786	. . . . . . . 8 ⊢ (𝑓 ∈ (Fil‘∪ 𝐽) → 𝑓 ∈ (fBas‘∪ 𝐽))
26	25	ad2antrl 728	. . . . . . 7 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (fBas‘∪ 𝐽))
27		simprrl 780	. . . . . . 7 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐿 ⊆ 𝑓)
28		fmss 23884	. . . . . . 7 ⊢ (((∪ 𝐾 ∈ 𝐾 ∧ 𝐿 ∈ (fBas‘∪ 𝐽) ∧ 𝑓 ∈ (fBas‘∪ 𝐽)) ∧ (𝐹:∪ 𝐽⟶∪ 𝐾 ∧ 𝐿 ⊆ 𝑓)) → ((∪ 𝐾 FilMap 𝐹)‘𝐿) ⊆ ((∪ 𝐾 FilMap 𝐹)‘𝑓))
29	19, 22, 26, 15, 27, 28	syl32anc 1380	. . . . . 6 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → ((∪ 𝐾 FilMap 𝐹)‘𝐿) ⊆ ((∪ 𝐾 FilMap 𝐹)‘𝑓))
30		fclsss2 23961	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ ((∪ 𝐾 FilMap 𝐹)‘𝐿) ∈ (Fil‘∪ 𝐾) ∧ ((∪ 𝐾 FilMap 𝐹)‘𝐿) ⊆ ((∪ 𝐾 FilMap 𝐹)‘𝑓)) → (𝐾 fClus ((∪ 𝐾 FilMap 𝐹)‘𝑓)) ⊆ (𝐾 fClus ((∪ 𝐾 FilMap 𝐹)‘𝐿)))
31	10, 24, 29, 30	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → (𝐾 fClus ((∪ 𝐾 FilMap 𝐹)‘𝑓)) ⊆ (𝐾 fClus ((∪ 𝐾 FilMap 𝐹)‘𝐿)))
32		fcfval 23971	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑓 ∈ (Fil‘∪ 𝐽) ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → ((𝐾 fClusf 𝑓)‘𝐹) = (𝐾 fClus ((∪ 𝐾 FilMap 𝐹)‘𝑓)))
33	10, 11, 15, 32	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → ((𝐾 fClusf 𝑓)‘𝐹) = (𝐾 fClus ((∪ 𝐾 FilMap 𝐹)‘𝑓)))
34		fcfval 23971	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐿 ∈ (Fil‘∪ 𝐽) ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → ((𝐾 fClusf 𝐿)‘𝐹) = (𝐾 fClus ((∪ 𝐾 FilMap 𝐹)‘𝐿)))
35	10, 20, 15, 34	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → ((𝐾 fClusf 𝐿)‘𝐹) = (𝐾 fClus ((∪ 𝐾 FilMap 𝐹)‘𝐿)))
36	31, 33, 35	3sstr4d 4014	. . . 4 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → ((𝐾 fClusf 𝑓)‘𝐹) ⊆ ((𝐾 fClusf 𝐿)‘𝐹))
37	17, 36	sstrd 3969	. . 3 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → ((𝐾 fLimf 𝑓)‘𝐹) ⊆ ((𝐾 fClusf 𝐿)‘𝐹))
38		simprrr 781	. . . 4 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐴 ∈ (𝐽 fLim 𝑓))
39		simpl3 1194	. . . 4 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
40		cnpflfi 23937	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹))
41	38, 39, 40	syl2anc 584	. . 3 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹))
42	37, 41	sseldd 3959	. 2 ⊢ (((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑓 ∈ (Fil‘∪ 𝐽) ∧ (𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝐿)‘𝐹))
43	7, 42	rexlimddv 3147	1 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝐿)‘𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∃wrex 3060   ⊆ wss 3926  ∪ cuni 4883  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  fBascfbas 21303  Topctop 22831  TopOnctopon 22848   CnP ccnp 23163  Filcfil 23783   FilMap cfm 23871   fLim cflim 23872   fLimf cflf 23873   fClus cfcls 23874   fClusf cfcf 23875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-1o 8480  df-2o 8481  df-map 8842  df-en 8960  df-fin 8963  df-fi 9423  df-fbas 21312  df-fg 21313  df-top 22832  df-topon 22849  df-cld 22957  df-ntr 22958  df-cls 22959  df-nei 23036  df-cnp 23166  df-fil 23784  df-fm 23876  df-flim 23877  df-flf 23878  df-fcls 23879  df-fcf 23880

This theorem is referenced by:  cnpfcf  23979