Theorem infil 23892

Description: The intersection of two filters is a filter. Use fiint 9394 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 4258	. . . 4 ⊢ (𝐹 ∩ 𝐺) ⊆ 𝐹
2		filsspw 23880	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
3	2	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
4	1, 3	sstrid 4020	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹 ∩ 𝐺) ⊆ 𝒫 𝑋)
5		0nelfil 23878	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
6	5	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
7		elinel1 4224	. . . 4 ⊢ (∅ ∈ (𝐹 ∩ 𝐺) → ∅ ∈ 𝐹)
8	6, 7	nsyl 140	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (𝐹 ∩ 𝐺))
9		filtop 23884	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
10	9	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋 ∈ 𝐹)
11		filtop 23884	. . . . 5 ⊢ (𝐺 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐺)
12	11	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋 ∈ 𝐺)
13	10, 12	elind 4223	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋 ∈ (𝐹 ∩ 𝐺))
14	4, 8, 13	3jca 1128	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ((𝐹 ∩ 𝐺) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ (𝐹 ∩ 𝐺) ∧ 𝑋 ∈ (𝐹 ∩ 𝐺)))
15		simpll 766	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝐹 ∈ (Fil‘𝑋))
16		simpr2 1195	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ (𝐹 ∩ 𝐺))
17		elinel1 4224	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐹 ∩ 𝐺) → 𝑦 ∈ 𝐹)
18	16, 17	syl 17	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ 𝐹)
19		simpr1 1194	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝒫 𝑋)
20	19	elpwid 4631	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ⊆ 𝑋)
21		simpr3 1196	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ 𝑥)
22		filss 23882	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐹)
23	15, 18, 20, 21, 22	syl13anc 1372	. . . . . . 7 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐹)
24		simplr 768	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝐺 ∈ (Fil‘𝑋))
25		elinel2 4225	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐹 ∩ 𝐺) → 𝑦 ∈ 𝐺)
26	16, 25	syl 17	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ 𝐺)
27		filss 23882	. . . . . . . 8 ⊢ ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐺 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐺)
28	24, 26, 20, 21, 27	syl13anc 1372	. . . . . . 7 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐺)
29	23, 28	elind 4223	. . . . . 6 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ (𝐹 ∩ 𝐺))
30	29	3exp2 1354	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝒫 𝑋 → (𝑦 ∈ (𝐹 ∩ 𝐺) → (𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺)))))
31	30	imp 406	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (𝑦 ∈ (𝐹 ∩ 𝐺) → (𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺))))
32	31	rexlimdv 3159	. . 3 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ (𝐹 ∩ 𝐺)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺)))
33	32	ralrimiva 3152	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ (𝐹 ∩ 𝐺)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺)))
34		simpl 482	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
35		elinel1 4224	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 ∩ 𝐺) → 𝑥 ∈ 𝐹)
36	35, 17	anim12i 612	. . . . 5 ⊢ ((𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺)) → (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹))
37		filin 23883	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ∩ 𝑦) ∈ 𝐹)
38	37	3expb 1120	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ∈ 𝐹)
39	34, 36, 38	syl2an 595	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺))) → (𝑥 ∩ 𝑦) ∈ 𝐹)
40		simpr 484	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐺 ∈ (Fil‘𝑋))
41		elinel2 4225	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 ∩ 𝐺) → 𝑥 ∈ 𝐺)
42	41, 25	anim12i 612	. . . . 5 ⊢ ((𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺)) → (𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺))
43		filin 23883	. . . . . 6 ⊢ ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺) → (𝑥 ∩ 𝑦) ∈ 𝐺)
44	43	3expb 1120	. . . . 5 ⊢ ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺)) → (𝑥 ∩ 𝑦) ∈ 𝐺)
45	40, 42, 44	syl2an 595	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺))) → (𝑥 ∩ 𝑦) ∈ 𝐺)
46	39, 45	elind 4223	. . 3 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺))) → (𝑥 ∩ 𝑦) ∈ (𝐹 ∩ 𝐺))
47	46	ralrimivva 3208	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ (𝐹 ∩ 𝐺)∀𝑦 ∈ (𝐹 ∩ 𝐺)(𝑥 ∩ 𝑦) ∈ (𝐹 ∩ 𝐺))
48		isfil2 23885	. 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 1343	1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹 ∩ 𝐺) ∈ (Fil‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  ‘cfv 6573  Filcfil 23874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-fbas 21384  df-fil 23875

This theorem is referenced by: (None)