Theorem cnpfcfi 24267

Description: Lemma for cnpfcf 24268. 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 1155	. . 3 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fClus 𝐿))
2		eqid 2762	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	fclsfil 24237	. . . . 5 ⊢ (𝐴 ∈ (𝐽 fClus 𝐿) → 𝐿 ∈ (Fil‘∪ 𝐽))
4	3	3ad2ant2 1152	. . . 4 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐿 ∈ (Fil‘∪ 𝐽))
5		fclsfnflim 24254	. . . 4 ⊢ (𝐿 ∈ (Fil‘∪ 𝐽) → (𝐴 ∈ (𝐽 fClus 𝐿) ↔ ∃𝑓 ∈ (Fil‘∪ 𝐽)(𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓))))
6	4, 5	syl 18	. . 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 235	. 2 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∃𝑓 ∈ (Fil‘∪ 𝐽)(𝐿 ⊆ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))
8		simpl1 1210	. . . . . 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 23144	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
10	8, 9	sylib 221	. . . . 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 783	. . . . 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 2762	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
13	2, 12	cnpf 23473	. . . . . . 7 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹:∪ 𝐽⟶∪ 𝐾)
14	13	3ad2ant3 1153	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:∪ 𝐽⟶∪ 𝐾)
15	14	adantr 486	. . . . 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 24265	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑓 ∈ (Fil‘∪ 𝐽) ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → ((𝐾 fLimf 𝑓)‘𝐹) ⊆ ((𝐾 fClusf 𝑓)‘𝐹))
17	10, 11, 15, 16	syl3anc 1398	. . . 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 23132	. . . . . . . 8 ⊢ (𝐾 ∈ Top → ∪ 𝐾 ∈ 𝐾)
19	8, 18	syl 18	. . . . . . 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 486	. . . . . . . 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 24075	. . . . . . . 8 ⊢ (𝐿 ∈ (Fil‘∪ 𝐽) → 𝐿 ∈ (fBas‘∪ 𝐽))
22	20, 21	syl 18	. . . . . . 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 24171	. . . . . . 7 ⊢ ((∪ 𝐾 ∈ 𝐾 ∧ 𝐿 ∈ (fBas‘∪ 𝐽) ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → ((∪ 𝐾 FilMap 𝐹)‘𝐿) ∈ (Fil‘∪ 𝐾))
24	19, 22, 15, 23	syl3anc 1398	. . . . . 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 24075	. . . . . . . 8 ⊢ (𝑓 ∈ (Fil‘∪ 𝐽) → 𝑓 ∈ (fBas‘∪ 𝐽))
26	25	ad2antrl 741	. . . . . . 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 793	. . . . . . 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 24173	. . . . . . 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 1405	. . . . . 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 24250	. . . . . 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 1398	. . . . 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 24260	. . . . . 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 1398	. . . . 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 24260	. . . . . 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 1398	. . . . 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 3989	. . . 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 3944	. . 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 794	. . . 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 1212	. . . 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 24226	. . . 4 ⊢ ((𝐴 ∈ (𝐽 fLim 𝑓) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹))
41	38, 39, 40	syl2anc 596	. . 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 3935	. 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 3171	1 ⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝐿)‘𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3088   ⊆ wss 3902  ∪ cuni 4870  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416  fBascfbas 21574  Topctop 23119  TopOnctopon 23136   CnP ccnp 23451  Filcfil 24072   FilMap cfm 24160   fLim cflim 24161   fLimf cflf 24162   fClus cfcls 24163   fClusf cfcf 24164

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-1o 8458  df-2o 8459  df-map 8831  df-en 8956  df-fin 8959  df-fi 9384  df-fbas 21583  df-fg 21584  df-top 23120  df-topon 23137  df-cld 23245  df-ntr 23246  df-cls 23247  df-nei 23324  df-cnp 23454  df-fil 24073  df-fm 24165  df-flim 24166  df-flf 24167  df-fcls 24168  df-fcf 24169

This theorem is used by:  cnpfcf  24268