Theorem isfcls 24131

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 473	. 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 6910	. . . . . . . 8 ⊢ (Fil‘𝑋) ⊆ ∪ ran Fil
3	2	sseli 3941	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ ∪ ran Fil)
4		filunibas 24003	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
5	4	eqcomd 2775	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 = ∪ 𝐹)
6	3, 5	jca 520	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹))
7		filunirn 24004	. . . . . . 7 ⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹))
8		fveq2 6879	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝐹 → (Fil‘𝑋) = (Fil‘∪ 𝐹))
9	8	eleq2d 2855	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝐹 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘∪ 𝐹)))
10	9	biimparc 484	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑋 = ∪ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
11	7, 10	sylanb 592	. . . . . 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 634	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) ↔ (𝐽 ∈ Top ∧ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹)))
14	13	anbi1i 635	. . 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 1103	. . 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 473	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) ↔ (𝐽 ∈ Top ∧ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹)))
17	16	anbi1i 635	. . 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 24063	. . . 4 ⊢ fClus = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑥 ∈ 𝑓 ((cls‘𝑗)‘𝑥), ∅))
20	19	elmpocl 7649	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil))
21		fclsval.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
22	21	fclsval 24130	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐹)) → (𝐽 fClus 𝐹) = if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅))
23	7, 22	sylan2b 605	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐽 fClus 𝐹) = if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅))
24	23	eleq2d 2855	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ 𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅)))
25		n0i 4301	. . . . . . 7 ⊢ (𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) → ¬ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∅)
26		iffalse 4498	. . . . . . 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 571	. . . 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 4495	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝐹 → if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠))
31	30	adantl 486	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠))
32	31	eleq2d 2855	. . . . . 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 3484	. . . . . . . 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 23984	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → 𝐹 ≠ ∅)
36	7, 35	sylbi 220	. . . . . . . . . 10 ⊢ (𝐹 ∈ ∪ ran Fil → 𝐹 ≠ ∅)
37	36	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → 𝐹 ≠ ∅)
38		r19.2z 4462	. . . . . . . . . 10 ⊢ ((𝐹 ≠ ∅ ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))
39	38	ex 417	. . . . . . . . 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 3484	. . . . . . . . 9 ⊢ (𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V)
42	41	rexlimivw 3168	. . . . . . . 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 4962	. . . . . . . 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 589	. . . 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 833	. 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)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  Vcvv 3463  ∅c0 4294  ifcif 4489  ∪ cuni 4873  ∩ ciin 4958  ran crn 5660  ‘cfv 6534  (class class class)co 7408  Topctop 23015  clsccl 23140  Filcfil 23967   fClus cfcls 24058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  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-iota 6490  df-fun 6536  df-fn 6537  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-fbas 21484  df-fil 23968  df-fcls 24063

This theorem is referenced by:  fclsfil  24132  fclstop  24133  isfcls2  24135  fclssscls  24140  flimfcls  24148