Theorem fclsval 23951

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 6863	. . . . 5 ⊢ (Fil‘𝑌) ⊆ ∪ ran Fil
3	2	sseli 3918	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑌) → 𝐹 ∈ ∪ ran Fil)
4	3	adantl 481	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐹 ∈ ∪ ran Fil)
5		filn0 23805	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑌) → 𝐹 ≠ ∅)
6	5	adantl 481	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → 𝐹 ≠ ∅)
7		fvex 6845	. . . . . 6 ⊢ ((cls‘𝐽)‘𝑡) ∈ V
8	7	rgenw 3056	. . . . 5 ⊢ ∀𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡) ∈ V
9		iinexg 5283	. . . . 5 ⊢ ((𝐹 ≠ ∅ ∧ ∀𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡) ∈ V) → ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡) ∈ V)
10	6, 8, 9	sylancl 587	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡) ∈ V)
11		0ex 5242	. . . 4 ⊢ ∅ ∈ V
12		ifcl 4513	. . . 4 ⊢ ((∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡) ∈ V ∧ ∅ ∈ V) → if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V)
13	10, 11, 12	sylancl 587	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅) ∈ V)
14		unieq 4862	. . . . . . 7 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
15		fclsval.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
16	14, 15	eqtr4di 2790	. . . . . 6 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
17		unieq 4862	. . . . . 6 ⊢ (𝑓 = 𝐹 → ∪ 𝑓 = ∪ 𝐹)
18	16, 17	eqeqan12d 2751	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → (∪ 𝑗 = ∪ 𝑓 ↔ 𝑋 = ∪ 𝐹))
19		iineq1 4952	. . . . . . 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 767	. . . . . . . . 9 ⊢ (((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) ∧ 𝑡 ∈ 𝐹) → 𝑗 = 𝐽)
22	21	fveq2d 6836	. . . . . . . 8 ⊢ (((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) ∧ 𝑡 ∈ 𝐹) → (cls‘𝑗) = (cls‘𝐽))
23	22	fveq1d 6834	. . . . . . 7 ⊢ (((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) ∧ 𝑡 ∈ 𝐹) → ((cls‘𝑗)‘𝑡) = ((cls‘𝐽)‘𝑡))
24	23	iineq2dv 4960	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∩ 𝑡 ∈ 𝐹 ((cls‘𝑗)‘𝑡) = ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡))
25	20, 24	eqtrd 2772	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → ∩ 𝑡 ∈ 𝑓 ((cls‘𝑗)‘𝑡) = ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡))
26	18, 25	ifbieq1d 4492	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑓 = 𝐹) → if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑡 ∈ 𝑓 ((cls‘𝑗)‘𝑡), ∅) = if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))
27		df-fcls 23884	. . . 4 ⊢ fClus = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑡 ∈ 𝑓 ((cls‘𝑗)‘𝑡), ∅))
28	26, 27	ovmpoga 7512	. . 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 1374	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝐽 fClus 𝐹) = if(𝑋 = ∪ 𝐹, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))
30		filunibas 23824	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑌) → ∪ 𝐹 = 𝑌)
31	30	eqeq2d 2748	. . . 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 4491	. 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 2772	1 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑌, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3430  ∅c0 4274  ifcif 4467  ∪ cuni 4851  ∩ ciin 4935  ran crn 5623  ‘cfv 6490  (class class class)co 7358  Topctop 22836  clsccl 22961  Filcfil 23788   fClus cfcls 23879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-fbas 21308  df-fil 23789  df-fcls 23884

This theorem is referenced by:  isfcls  23952  fclscmpi  23972