Theorem cnpfcfi 23099

Description: Lemma for cnpfcf 23100. 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 1135	. . 3 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fClus 𝐿))
2		eqid 2738	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	fclsfil 23069	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fClus 𝐿) → 𝐿 ∈ (Fil‘∪ 𝐽))
4	3	3ad2ant2 1132	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐿 ∈ (Fil‘∪ 𝐽))
5		fclsfnflim 23086	. . . 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 231	. 2 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∃𝑓 ∈ (Fil‘∪ 𝐽)(𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))
8		simpl1 1189	. . . . . 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 21975	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
10	8, 9	sylib 217	. . . . 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 767	. . . . 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 2738	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
13	2, 12	cnpf 22306	. . . . . . 7 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹:∪ 𝐽⟶∪ 𝐾)
14	13	3ad2ant3 1133	. . . . . 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 23097	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑓 ∈ (Fil‘∪ 𝐽) ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → ((𝐾 fLimf 𝑓)‘𝐹) ⊆ ((𝐾 fClusf 𝑓)‘𝐹))
17	10, 11, 15, 16	syl3anc 1369	. . . 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 21963	. . . . . . . 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 22907	. . . . . . . 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 23003	. . . . . . 7 ⊢ ((∪ 𝐾 ∈ 𝐾 ∧ 𝐿 ∈ (fBas‘∪ 𝐽) ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → ((∪ 𝐾 FilMap 𝐹)‘𝐿) ∈ (Fil‘∪ 𝐾))
24	19, 22, 15, 23	syl3anc 1369	. . . . . 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 22907	. . . . . . . 8 ⊢ (𝑓 ∈ (Fil‘∪ 𝐽) → 𝑓 ∈ (fBas‘∪ 𝐽))
26	25	ad2antrl 724	. . . . . . 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 777	. . . . . . 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 23005	. . . . . . 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 1376	. . . . . 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 23082	. . . . . 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 1369	. . . . 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 23092	. . . . . 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 1369	. . . . 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 23092	. . . . . 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 1369	. . . . 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 3964	. . . 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 3927	. . 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 778	. . . 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 1191	. . . 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 23058	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹))
41	38, 39, 40	syl2anc 583	. . 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 3918	. 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 3219	1 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝐿)‘𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∃wrex 3064   ⊆ wss 3883  ∪ cuni 4836  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  fBascfbas 20498  Topctop 21950  TopOnctopon 21967   CnP ccnp 22284  Filcfil 22904   FilMap cfm 22992   fLim cflim 22993   fLimf cflf 22994   fClus cfcls 22995   fClusf cfcf 22996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-fin 8695  df-fi 9100  df-fbas 20507  df-fg 20508  df-top 21951  df-topon 21968  df-cld 22078  df-ntr 22079  df-cls 22080  df-nei 22157  df-cnp 22287  df-fil 22905  df-fm 22997  df-flim 22998  df-flf 22999  df-fcls 23000  df-fcf 23001

This theorem is referenced by:  cnpfcf  23100