Theorem isfcf 23385

Description: The property of being a cluster point of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Assertion

Ref	Expression
isfcf	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))))

Distinct variable groups:   𝐴,𝑜   𝑜,𝑠,𝐽   𝑜,𝐿,𝑠   𝑜,𝐹,𝑠   𝑜,𝑋,𝑠   𝑜,𝑌,𝑠

Allowed substitution hint:   𝐴(𝑠)

Proof of Theorem isfcf

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fcfval 23384	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))
2	1	eleq2d 2823	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ 𝐴 ∈ (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿))))
3		simp1 1136	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4		toponmax 22275	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
5		filfbas 23199	. . . 4 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
6		id 22	. . . 4 ⊢ (𝐹:𝑌⟶𝑋 → 𝐹:𝑌⟶𝑋)
7		fmfil 23295	. . . 4 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
8	4, 5, 6, 7	syl3an 1160	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
9		fclsopn 23365	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅))))
10	3, 8, 9	syl2anc 584	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅))))
11		simpll1 1212	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → 𝐽 ∈ (TopOn‘𝑋))
12	11, 4	syl 17	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → 𝑋 ∈ 𝐽)
13		simpll2 1213	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → 𝐿 ∈ (Fil‘𝑌))
14	13, 5	syl 17	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → 𝐿 ∈ (fBas‘𝑌))
15		simpll3 1214	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → 𝐹:𝑌⟶𝑋)
16		simpl2 1192	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → 𝐿 ∈ (Fil‘𝑌))
17		fgfil 23226	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ (Fil‘𝑌) → (𝑌filGen𝐿) = 𝐿)
18	16, 17	syl 17	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑌filGen𝐿) = 𝐿)
19	18	eleq2d 2823	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑠 ∈ (𝑌filGen𝐿) ↔ 𝑠 ∈ 𝐿))
20	19	biimpar 478	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → 𝑠 ∈ (𝑌filGen𝐿))
21		eqid 2736	. . . . . . . . . 10 ⊢ (𝑌filGen𝐿) = (𝑌filGen𝐿)
22	21	imaelfm 23302	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑠 ∈ (𝑌filGen𝐿)) → (𝐹 “ 𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿))
23	12, 14, 15, 20, 22	syl31anc 1373	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → (𝐹 “ 𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿))
24		ineq2 4166	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹 “ 𝑠) → (𝑜 ∩ 𝑥) = (𝑜 ∩ (𝐹 “ 𝑠)))
25	24	neeq1d 3003	. . . . . . . . 9 ⊢ (𝑥 = (𝐹 “ 𝑠) → ((𝑜 ∩ 𝑥) ≠ ∅ ↔ (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))
26	25	rspcv 3577	. . . . . . . 8 ⊢ ((𝐹 “ 𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅ → (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))
27	23, 26	syl 17	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅ → (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))
28	27	ralrimdva 3151	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅ → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))
29		elfm 23298	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑥)))
30	4, 5, 6, 29	syl3an 1160	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑥)))
31	30	adantr 481	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑥)))
32	31	simplbda 500	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑥)
33		r19.29r 3119	. . . . . . . . . 10 ⊢ ((∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑥 ∧ ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) → ∃𝑠 ∈ 𝐿 ((𝐹 “ 𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))
34		sslin 4194	. . . . . . . . . . . 12 ⊢ ((𝐹 “ 𝑠) ⊆ 𝑥 → (𝑜 ∩ (𝐹 “ 𝑠)) ⊆ (𝑜 ∩ 𝑥))
35		ssn0 4360	. . . . . . . . . . . 12 ⊢ (((𝑜 ∩ (𝐹 “ 𝑠)) ⊆ (𝑜 ∩ 𝑥) ∧ (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) → (𝑜 ∩ 𝑥) ≠ ∅)
36	34, 35	sylan 580	. . . . . . . . . . 11 ⊢ (((𝐹 “ 𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) → (𝑜 ∩ 𝑥) ≠ ∅)
37	36	rexlimivw 3148	. . . . . . . . . 10 ⊢ (∃𝑠 ∈ 𝐿 ((𝐹 “ 𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) → (𝑜 ∩ 𝑥) ≠ ∅)
38	33, 37	syl 17	. . . . . . . . 9 ⊢ ((∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑥 ∧ ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) → (𝑜 ∩ 𝑥) ≠ ∅)
39	38	ex 413	. . . . . . . 8 ⊢ (∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑥 → (∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑜 ∩ 𝑥) ≠ ∅))
40	32, 39	syl 17	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) → (∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑜 ∩ 𝑥) ≠ ∅))
41	40	ralrimdva 3151	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → (∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅ → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅))
42	28, 41	impbid 211	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅ ↔ ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))
43	42	imbi2d 340	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → ((𝐴 ∈ 𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅) ↔ (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅)))
44	43	ralbidva 3172	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅)))
45	44	anbi2d 629	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))))
46	2, 10, 45	3bitrd 304	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282   “ cima 5636  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  fBascfbas 20784  filGencfg 20785  TopOnctopon 22259  Filcfil 23196   FilMap cfm 23284   fClus cfcls 23287   fClusf cfcf 23288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-fbas 20793  df-fg 20794  df-top 22243  df-topon 22260  df-cld 22370  df-ntr 22371  df-cls 22372  df-fil 23197  df-fm 23289  df-fcls 23292  df-fcf 23293

This theorem is referenced by:  fcfnei  23386