Theorem isfcls 24203

Description: A cluster point of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Hypothesis

Ref	Expression
fclsval.x	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
isfcls	⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))

Distinct variable groups:   𝐴,𝑠   𝐹,𝑠   𝑋,𝑠   𝐽,𝑠

Proof of Theorem isfcls

Dummy variables 𝑓 𝑗 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		anass 474	. 2 ⊢ ((((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ (𝑋 = ∪ 𝐹 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
2		fvssunirn 6919	. . . . . . . 8 ⊢ (Fil‘𝑋) ⊆ ∪ ran Fil
3	2	sseli 3936	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ ∪ ran Fil)
4		filunibas 24075	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
5	4	eqcomd 2772	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 = ∪ 𝐹)
6	3, 5	jca 521	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹))
7		filunirn 24076	. . . . . . 7 ⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹))
8		fveq2 6888	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝐹 → (Fil‘𝑋) = (Fil‘∪ 𝐹))
9	8	eleq2d 2852	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝐹 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘∪ 𝐹)))
10	9	biimparc 485	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑋 = ∪ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
11	7, 10	sylanb 593	. . . . . 6 ⊢ ((𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
12	6, 11	impbii 212	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹))
13	12	anbi2i 635	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) ↔ (𝐽 ∈ Top ∧ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹)))
14	13	anbi1i 636	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ ((𝐽 ∈ Top ∧ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹)) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
15		df-3an 1105	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
16		anass 474	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) ↔ (𝐽 ∈ Top ∧ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹)))
17	16	anbi1i 636	. . 3 ⊢ ((((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ ((𝐽 ∈ Top ∧ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹)) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
18	14, 15, 17	3bitr4i 306	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
19		df-fcls 24135	. . . 4 ⊢ fClus = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑥 ∈ 𝑓 ((cls‘𝑗)‘𝑥), ∅))
20	19	elmpocl 7664	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil))
21		fclsval.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
22	21	fclsval 24202	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐹)) → (𝐽 fClus 𝐹) = if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅))
23	7, 22	sylan2b 606	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐽 fClus 𝐹) = if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅))
24	23	eleq2d 2852	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ 𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅)))
25		n0i 4296	. . . . . . 7 ⊢ (𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) → ¬ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∅)
26		iffalse 4501	. . . . . . 7 ⊢ (¬ 𝑋 = ∪ 𝐹 → if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∅)
27	25, 26	nsyl2 142	. . . . . 6 ⊢ (𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) → 𝑋 = ∪ 𝐹)
28	27	a1i 11	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) → 𝑋 = ∪ 𝐹))
29	28	pm4.71rd 572	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) ↔ (𝑋 = ∪ 𝐹 ∧ 𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅))))
30		iftrue 4498	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝐹 → if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠))
31	30	adantl 487	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠))
32	31	eleq2d 2852	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) ↔ 𝐴 ∈ ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠)))
33		elex 3479	. . . . . . . 8 ⊢ (𝐴 ∈ ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V)
34	33	a1i 11	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (𝐴 ∈ ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V))
35		filn0 24056	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → 𝐹 ≠ ∅)
36	7, 35	sylbi 220	. . . . . . . . . 10 ⊢ (𝐹 ∈ ∪ ran Fil → 𝐹 ≠ ∅)
37	36	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → 𝐹 ≠ ∅)
38		r19.2z 4465	. . . . . . . . . 10 ⊢ ((𝐹 ≠ ∅ ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))
39	38	ex 418	. . . . . . . . 9 ⊢ (𝐹 ≠ ∅ → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
40	37, 39	syl 18	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
41		elex 3479	. . . . . . . . 9 ⊢ (𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V)
42	41	rexlimivw 3165	. . . . . . . 8 ⊢ (∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V)
43	40, 42	syl6 36	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V))
44		eliin 4966	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
45	44	a1i 11	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (𝐴 ∈ V → (𝐴 ∈ ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
46	34, 43, 45	pm5.21ndd 382	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (𝐴 ∈ ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
47	32, 46	bitrd 282	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
48	47	pm5.32da 590	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → ((𝑋 = ∪ 𝐹 ∧ 𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅)) ↔ (𝑋 = ∪ 𝐹 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
49	24, 29, 48	3bitrd 308	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝑋 = ∪ 𝐹 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
50	20, 49	biadanii 834	. 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ (𝑋 = ∪ 𝐹 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
51	1, 18, 50	3bitr4ri 307	1 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∀wral 3082  ∃wrex 3092  Vcvv 3458  ∅c0 4289  ifcif 4492  ∪ cuni 4877  ∩ ciin 4962  ran crn 5667  ‘cfv 6543  (class class class)co 7423  Topctop 23087  clsccl 23212  Filcfil 24039   fClus cfcls 24130

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iin 4964  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-fbas 21556  df-fil 24040  df-fcls 24135

This theorem is used by:  fclsfil  24204  fclstop  24205  isfcls2  24207  fclssscls  24212  flimfcls  24220