Theorem isfcls 23068

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 468	. 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 6785	. . . . . . . 8 ⊢ (Fil‘𝑋) ⊆ ∪ ran Fil
3	2	sseli 3913	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ ∪ ran Fil)
4		filunibas 22940	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
5	4	eqcomd 2744	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 = ∪ 𝐹)
6	3, 5	jca 511	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹))
7		filunirn 22941	. . . . . . 7 ⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹))
8		fveq2 6756	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝐹 → (Fil‘𝑋) = (Fil‘∪ 𝐹))
9	8	eleq2d 2824	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝐹 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘∪ 𝐹)))
10	9	biimparc 479	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘∪ 𝐹) ∧ 𝑋 = ∪ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
11	7, 10	sylanb 580	. . . . . 6 ⊢ ((𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
12	6, 11	impbii 208	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹))
13	12	anbi2i 622	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋)) ↔ (𝐽 ∈ Top ∧ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹)))
14	13	anbi1i 623	. . 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 1087	. . 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 468	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) ↔ (𝐽 ∈ Top ∧ (𝐹 ∈ ∪ ran Fil ∧ 𝑋 = ∪ 𝐹)))
17	16	anbi1i 623	. . 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 302	. 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 23000	. . . 4 ⊢ fClus = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑥 ∈ 𝑓 ((cls‘𝑗)‘𝑥), ∅))
20	19	elmpocl 7489	. . 3 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil))
21		fclsval.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
22	21	fclsval 23067	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘∪ 𝐹)) → (𝐽 fClus 𝐹) = if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅))
23	7, 22	sylan2b 593	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐽 fClus 𝐹) = if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅))
24	23	eleq2d 2824	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ 𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅)))
25		n0i 4264	. . . . . . 7 ⊢ (𝐴 ∈ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) → ¬ if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∅)
26		iffalse 4465	. . . . . . 7 ⊢ (¬ 𝑋 = ∪ 𝐹 → if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∅)
27	25, 26	nsyl2 141	. . . . . 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 562	. . . 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 4462	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝐹 → if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠))
31	30	adantl 481	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → if(𝑋 = ∪ 𝐹, ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠), ∅) = ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠))
32	31	eleq2d 2824	. . . . . 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 3440	. . . . . . . 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 22921	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘∪ 𝐹) → 𝐹 ≠ ∅)
36	7, 35	sylbi 216	. . . . . . . . . 10 ⊢ (𝐹 ∈ ∪ ran Fil → 𝐹 ≠ ∅)
37	36	ad2antlr 723	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → 𝐹 ≠ ∅)
38		r19.2z 4422	. . . . . . . . . 10 ⊢ ((𝐹 ≠ ∅ ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))
39	38	ex 412	. . . . . . . . 9 ⊢ (𝐹 ≠ ∅ → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
40	37, 39	syl 17	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
41		elex 3440	. . . . . . . . 9 ⊢ (𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V)
42	41	rexlimivw 3210	. . . . . . . 8 ⊢ (∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V)
43	40, 42	syl6 35	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ V))
44		eliin 4926	. . . . . . . 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 380	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ 𝑋 = ∪ 𝐹) → (𝐴 ∈ ∩ 𝑠 ∈ 𝐹 ((cls‘𝐽)‘𝑠) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
47	32, 46	bitrd 278	. . . . 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 578	. . . 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 304	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝑋 = ∪ 𝐹 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
50	20, 49	biadanii 818	. 2 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) ∧ (𝑋 = ∪ 𝐹 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))))
51	1, 18, 50	3bitr4ri 303	1 ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422  ∅c0 4253  ifcif 4456  ∪ cuni 4836  ∩ ciin 4922  ran crn 5581  ‘cfv 6418  (class class class)co 7255  Topctop 21950  clsccl 22077  Filcfil 22904   fClus cfcls 22995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-fbas 20507  df-fil 22905  df-fcls 23000

This theorem is referenced by:  fclsfil  23069  fclstop  23070  isfcls2  23072  fclssscls  23077  flimfcls  23085