Theorem cnpfcfi 24386

Description: Lemma for cnpfcf 24387. 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 2763	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	fclsfil 24356	. . . . 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 24373	. . . 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 23263	. . . . . 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 2763	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
13	2, 12	cnpf 23592	. . . . . . 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 24384	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑓 ∈ (Fil‘∪ 𝐽) ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → ((𝐾 fLimf 𝑓)‘𝐹) ⊆ ((𝐾 fClusf 𝑓)‘𝐹))
17	10, 11, 15, 16	syl3anc 1399	. . . 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 23251	. . . . . . . 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 24194	. . . . . . . 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 24290	. . . . . . 7 ⊢ ((∪ 𝐾 ∈ 𝐾 ∧ 𝐿 ∈ (fBas‘∪ 𝐽) ∧ 𝐹:∪ 𝐽⟶∪ 𝐾) → ((∪ 𝐾 FilMap 𝐹)‘𝐿) ∈ (Fil‘∪ 𝐾))
24	19, 22, 15, 23	syl3anc 1399	. . . . . 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 24194	. . . . . . . 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 24292	. . . . . . 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 1406	. . . . . 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 24369	. . . . . 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 1399	. . . . 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 24379	. . . . . 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 1399	. . . . 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 24379	. . . . . 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 1399	. . . . 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 3968	. . . 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 3923	. . 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 24345	. . . 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 3914	. 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 3173	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 2147  ∃wrex 3090   ⊆ wss 3900  ∪ cuni 4870  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440  fBascfbas 21693  Topctop 23238  TopOnctopon 23255   CnP ccnp 23570  Filcfil 24191   FilMap cfm 24279   fLim cflim 24280   fLimf cflf 24281   fClus cfcls 24282   fClusf cfcf 24283

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-1o 8501  df-2o 8502  df-map 8874  df-en 8999  df-fin 9002  df-fi 9427  df-fbas 21702  df-fg 21703  df-top 23239  df-topon 23256  df-cld 23364  df-ntr 23365  df-cls 23366  df-nei 23443  df-cnp 23573  df-fil 24192  df-fm 24284  df-flim 24285  df-flf 24286  df-fcls 24287  df-fcf 24288

This theorem is used by:  cnpfcf  24387