Theorem fclsval 23905

Description: The set of all cluster points 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
fclsval	⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑌, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))

Distinct variable groups:   𝑡,𝐹   𝑡,𝐽

Allowed substitution hints:   𝑋(𝑡)   𝑌(𝑡)

Proof of Theorem fclsval

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

Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐽 ∈ Top)
2		fvssunirn 6924	. . . . 5 ⊢ (Fil‘𝑌) ⊆ ∪ ran Fil
3	2	sseli 3974	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑌) → 𝐹 ∈ ∪ ran Fil)
4	3	adantl 481	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐹 ∈ ∪ ran Fil)
5		filn0 23759	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑌) → 𝐹 ≠ ∅)
6	5	adantl 481	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐹 ≠ ∅)
7		fvex 6904	. . . . . 6 ⊢ ((cls‘𝐽)‘𝑡) ∈ V
8	7	rgenw 3061	. . . . 5 ⊢ ∀𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡) ∈ V
9		iinexg 5337	. . . . 5 ⊢ ((𝐹 ≠ ∅ ∧ ∀𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡) ∈ V) → ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡) ∈ V)
10	6, 8, 9	sylancl 585	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡) ∈ V)
11		0ex 5301	. . . 4 ⊢ ∅ ∈ V
12		ifcl 4569	. . . 4 ⊢ ((∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡) ∈ V ∧ ∅ ∈ V) → if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V)
13	10, 11, 12	sylancl 585	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V)
14		unieq 4914	. . . . . . 7 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
15		fclsval.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
16	14, 15	eqtr4di 2786	. . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
17		unieq 4914	. . . . . 6 ⊢ (𝑓 = 𝐹 → ∪ 𝑓 = ∪ 𝐹)
18	16, 17	eqeqan12d 2742	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (∪ 𝑗 = ∪ 𝑓 ↔ 𝑋 = ∪ 𝐹))
19		iineq1 5008	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ∩ 𝑡 ∈ 𝑓 ((cls‘𝑗)‘𝑡) = ∩ 𝑡 ∈ 𝐹 ((cls‘𝑗)‘𝑡))
20	19	adantl 481	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∩ 𝑡 ∈ 𝑓 ((cls‘𝑗)‘𝑡) = ∩ 𝑡 ∈ 𝐹 ((cls‘𝑗)‘𝑡))
21		simpll 766	. . . . . . . . 9 ⊢ (((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) ∧ 𝑡 ∈ 𝐹) → 𝑗 = 𝐽)
22	21	fveq2d 6895	. . . . . . . 8 ⊢ (((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) ∧ 𝑡 ∈ 𝐹) → (cls‘𝑗) = (cls‘𝐽))
23	22	fveq1d 6893	. . . . . . 7 ⊢ (((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) ∧ 𝑡 ∈ 𝐹) → ((cls‘𝑗)‘𝑡) = ((cls‘𝐽)‘𝑡))
24	23	iineq2dv 5016	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∩ 𝑡 ∈ 𝐹 ((cls‘𝑗)‘𝑡) = ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡))
25	20, 24	eqtrd 2768	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∩ 𝑡 ∈ 𝑓 ((cls‘𝑗)‘𝑡) = ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡))
26	18, 25	ifbieq1d 4548	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑡 ∈ 𝑓 ((cls‘𝑗)‘𝑡), ∅) = if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))
27		df-fcls 23838	. . . 4 ⊢ fClus = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑡 ∈ 𝑓 ((cls‘𝑗)‘𝑡), ∅))
28	26, 27	ovmpoga 7569	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V) → (𝐽 fClus 𝐹) = if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))
29	1, 4, 13, 28	syl3anc 1369	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝐽 fClus 𝐹) = if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))
30		filunibas 23778	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑌) → ∪ 𝐹 = 𝑌)
31	30	eqeq2d 2739	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑌) → (𝑋 = ∪ 𝐹 ↔ 𝑋 = 𝑌))
32	31	adantl 481	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝑋 = ∪ 𝐹 ↔ 𝑋 = 𝑌))
33	32	ifbid 4547	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅) = if(𝑋 = 𝑌, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))
34	29, 33	eqtrd 2768	1 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑌, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ≠ wne 2936  ∀wral 3057  Vcvv 3470  ∅c0 4318  ifcif 4524  ∪ cuni 4903  ∩ ciin 4992  ran crn 5673  ‘cfv 6542  (class class class)co 7414  Topctop 22788  clsccl 22915  Filcfil 23742   fClus cfcls 23833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-fbas 21269  df-fil 23743  df-fcls 23838

This theorem is referenced by:  isfcls  23906  fclscmpi  23926

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