Theorem isfcf 23921

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 23920	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))
2	1	eleq2d 2814	. 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 22813	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
5		filfbas 23735	. . . 4 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
6		id 22	. . . 4 ⊢ (𝐹:𝑌⟶𝑋 → 𝐹:𝑌⟶𝑋)
7		fmfil 23831	. . . 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 23901	. . 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 1213	. . . . . . . . . 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 1214	. . . . . . . . . 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 1215	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → 𝐹:𝑌⟶𝑋)
16		simpl2 1193	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → 𝐿 ∈ (Fil‘𝑌))
17		fgfil 23762	. . . . . . . . . . . 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 2814	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑠 ∈ (𝑌filGen𝐿) ↔ 𝑠 ∈ 𝐿))
20	19	biimpar 477	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → 𝑠 ∈ (𝑌filGen𝐿))
21		eqid 2729	. . . . . . . . . 10 ⊢ (𝑌filGen𝐿) = (𝑌filGen𝐿)
22	21	imaelfm 23838	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑠 ∈ (𝑌filGen𝐿)) → (𝐹 “ 𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿))
23	12, 14, 15, 20, 22	syl31anc 1375	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → (𝐹 “ 𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿))
24		ineq2 4177	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹 “ 𝑠) → (𝑜 ∩ 𝑥) = (𝑜 ∩ (𝐹 “ 𝑠)))
25	24	neeq1d 2984	. . . . . . . . 9 ⊢ (𝑥 = (𝐹 “ 𝑠) → ((𝑜 ∩ 𝑥) ≠ ∅ ↔ (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))
26	25	rspcv 3584	. . . . . . . 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 3133	. . . . . 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 23834	. . . . . . . . . . 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 480	. . . . . . . . 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 499	. . . . . . . 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 3096	. . . . . . . . . 10 ⊢ ((∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑥 ∧ ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) → ∃𝑠 ∈ 𝐿 ((𝐹 “ 𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))
34		sslin 4206	. . . . . . . . . . . 12 ⊢ ((𝐹 “ 𝑠) ⊆ 𝑥 → (𝑜 ∩ (𝐹 “ 𝑠)) ⊆ (𝑜 ∩ 𝑥))
35		ssn0 4367	. . . . . . . . . . . 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 3130	. . . . . . . . . 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 412	. . . . . . . 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 3133	. . . . . 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 212	. . . . 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 3154	. . 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 630	. 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 305	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296   “ cima 5641  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  fBascfbas 21252  filGencfg 21253  TopOnctopon 22797  Filcfil 23732   FilMap cfm 23820   fClus cfcls 23823   fClusf cfcf 23824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-fbas 21261  df-fg 21262  df-top 22781  df-topon 22798  df-cld 22906  df-ntr 22907  df-cls 22908  df-fil 23733  df-fm 23825  df-fcls 23828  df-fcf 23829

This theorem is referenced by:  fcfnei  23922