Theorem infil 23988

Description: The intersection of two filters is a filter. Use fiint 9285 to extend this property to the intersection of a finite set of filters. Paragraph 3 of [BourbakiTop1] p. I.36. (Contributed by FL, 17-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
infil	⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹 ∩ 𝐺) ∈ (Fil‘𝑋))

Proof of Theorem infil

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 4197	. . . 4 ⊢ (𝐹 ∩ 𝐺) ⊆ 𝐹
2		filsspw 23976	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
3	2	adantr 485	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
4	1, 3	sstrid 3956	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹 ∩ 𝐺) ⊆ 𝒫 𝑋)
5		0nelfil 23974	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
6	5	adantr 485	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
7		elinel1 4162	. . . 4 ⊢ (∅ ∈ (𝐹 ∩ 𝐺) → ∅ ∈ 𝐹)
8	6, 7	nsyl 141	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (𝐹 ∩ 𝐺))
9		filtop 23980	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
10	9	adantr 485	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋 ∈ 𝐹)
11		filtop 23980	. . . . 5 ⊢ (𝐺 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐺)
12	11	adantl 486	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋 ∈ 𝐺)
13	10, 12	elind 4161	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋 ∈ (𝐹 ∩ 𝐺))
14	4, 8, 13	3jca 1144	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ((𝐹 ∩ 𝐺) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ (𝐹 ∩ 𝐺) ∧ 𝑋 ∈ (𝐹 ∩ 𝐺)))
15		simpll 778	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝐹 ∈ (Fil‘𝑋))
16		simpr2 1212	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ (𝐹 ∩ 𝐺))
17		elinel1 4162	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐹 ∩ 𝐺) → 𝑦 ∈ 𝐹)
18	16, 17	syl 18	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ 𝐹)
19		simpr1 1211	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝒫 𝑋)
20	19	elpwid 4576	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ⊆ 𝑋)
21		simpr3 1213	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ 𝑥)
22		filss 23978	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐹)
23	15, 18, 20, 21, 22	syl13anc 1397	. . . . . . 7 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐹)
24		simplr 780	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝐺 ∈ (Fil‘𝑋))
25		elinel2 4163	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐹 ∩ 𝐺) → 𝑦 ∈ 𝐺)
26	16, 25	syl 18	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ 𝐺)
27		filss 23978	. . . . . . . 8 ⊢ ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐺 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐺)
28	24, 26, 20, 21, 27	syl13anc 1397	. . . . . . 7 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐺)
29	23, 28	elind 4161	. . . . . 6 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ (𝐹 ∩ 𝐺))
30	29	3exp2 1371	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝒫 𝑋 → (𝑦 ∈ (𝐹 ∩ 𝐺) → (𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺)))))
31	30	imp 411	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (𝑦 ∈ (𝐹 ∩ 𝐺) → (𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺))))
32	31	rexlimdv 3170	. . 3 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ (𝐹 ∩ 𝐺)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺)))
33	32	ralrimiva 3163	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ (𝐹 ∩ 𝐺)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺)))
34		simpl 487	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
35		elinel1 4162	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 ∩ 𝐺) → 𝑥 ∈ 𝐹)
36	35, 17	anim12i 624	. . . . 5 ⊢ ((𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺)) → (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹))
37		filin 23979	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ∩ 𝑦) ∈ 𝐹)
38	37	3expb 1136	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ∈ 𝐹)
39	34, 36, 38	syl2an 607	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺))) → (𝑥 ∩ 𝑦) ∈ 𝐹)
40		simpr 489	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐺 ∈ (Fil‘𝑋))
41		elinel2 4163	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 ∩ 𝐺) → 𝑥 ∈ 𝐺)
42	41, 25	anim12i 624	. . . . 5 ⊢ ((𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺)) → (𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺))
43		filin 23979	. . . . . 6 ⊢ ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺) → (𝑥 ∩ 𝑦) ∈ 𝐺)
44	43	3expb 1136	. . . . 5 ⊢ ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺)) → (𝑥 ∩ 𝑦) ∈ 𝐺)
45	40, 42, 44	syl2an 607	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺))) → (𝑥 ∩ 𝑦) ∈ 𝐺)
46	39, 45	elind 4161	. . 3 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺))) → (𝑥 ∩ 𝑦) ∈ (𝐹 ∩ 𝐺))
47	46	ralrimivva 3214	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ (𝐹 ∩ 𝐺)∀𝑦 ∈ (𝐹 ∩ 𝐺)(𝑥 ∩ 𝑦) ∈ (𝐹 ∩ 𝐺))
48		isfil2 23981	. 2 ⊢ ((𝐹 ∩ 𝐺) ∈ (Fil‘𝑋) ↔ (((𝐹 ∩ 𝐺) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ (𝐹 ∩ 𝐺) ∧ 𝑋 ∈ (𝐹 ∩ 𝐺)) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ (𝐹 ∩ 𝐺)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺)) ∧ ∀𝑥 ∈ (𝐹 ∩ 𝐺)∀𝑦 ∈ (𝐹 ∩ 𝐺)(𝑥 ∩ 𝑦) ∈ (𝐹 ∩ 𝐺)))
49	14, 33, 47, 48	syl3anbrc 1360	1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹 ∩ 𝐺) ∈ (Fil‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1101   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  ‘cfv 6537  Filcfil 23970

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 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405

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 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fv 6545  df-fbas 21487  df-fil 23971

This theorem is referenced by: (None)