Theorem cnpfcfi 23955

Description: Lemma for cnpfcf 23956. 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 2731	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	fclsfil 23925	. . . . 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 23942	. . . 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 22833	. . . . . 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 2731	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
13	2, 12	cnpf 23162	. . . . . . 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 23953	. . . . 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 22821	. . . . . . . 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 23763	. . . . . . . 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 23859	. . . . . . 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 23763	. . . . . . . 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 23861	. . . . . . 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 23938	. . . . . 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 23948	. . . . . 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 23948	. . . . . 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 3985	. . . 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 3940	. . 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 23914	. . . 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 3930	. 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 3139	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 1541   ∈ wcel 2111  ∃wrex 3056   ⊆ wss 3897  ∪ cuni 4856  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  fBascfbas 21279  Topctop 22808  TopOnctopon 22825   CnP ccnp 23140  Filcfil 23760   FilMap cfm 23848   fLim cflim 23849   fLimf cflf 23850   fClus cfcls 23851   fClusf cfcf 23852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-1o 8385  df-2o 8386  df-map 8752  df-en 8870  df-fin 8873  df-fi 9295  df-fbas 21288  df-fg 21289  df-top 22809  df-topon 22826  df-cld 22934  df-ntr 22935  df-cls 22936  df-nei 23013  df-cnp 23143  df-fil 23761  df-fm 23853  df-flim 23854  df-flf 23855  df-fcls 23856  df-fcf 23857

This theorem is referenced by:  cnpfcf  23956