Theorem infil 23846

Description: The intersection of two filters is a filter. Use fiint 9227 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 4165	. . . 4 ⊢ (𝐹 ∩ 𝐺) ⊆ 𝐹
2		filsspw 23834	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
3	2	adantr 481	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
4	1, 3	sstrid 3926	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹 ∩ 𝐺) ⊆ 𝒫 𝑋)
5		0nelfil 23832	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
6	5	adantr 481	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
7		elinel1 4130	. . . 4 ⊢ (∅ ∈ (𝐹 ∩ 𝐺) → ∅ ∈ 𝐹)
8	6, 7	nsyl 140	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (𝐹 ∩ 𝐺))
9		filtop 23838	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
10	9	adantr 481	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋 ∈ 𝐹)
11		filtop 23838	. . . . 5 ⊢ (𝐺 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐺)
12	11	adantl 482	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋 ∈ 𝐺)
13	10, 12	elind 4129	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋 ∈ (𝐹 ∩ 𝐺))
14	4, 8, 13	3jca 1134	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ((𝐹 ∩ 𝐺) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ (𝐹 ∩ 𝐺) ∧ 𝑋 ∈ (𝐹 ∩ 𝐺)))
15		simpll 772	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝐹 ∈ (Fil‘𝑋))
16		simpr2 1202	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ (𝐹 ∩ 𝐺))
17		elinel1 4130	. . . . . . . . 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 1201	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝒫 𝑋)
20	19	elpwid 4538	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ⊆ 𝑋)
21		simpr3 1203	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ 𝑥)
22		filss 23836	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐹)
23	15, 18, 20, 21, 22	syl13anc 1380	. . . . . . 7 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐹)
24		simplr 774	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝐺 ∈ (Fil‘𝑋))
25		elinel2 4131	. . . . . . . . 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 23836	. . . . . . . 8 ⊢ ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐺 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐺)
28	24, 26, 20, 21, 27	syl13anc 1380	. . . . . . 7 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐺)
29	23, 28	elind 4129	. . . . . 6 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ (𝐹 ∩ 𝐺))
30	29	3exp2 1361	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝒫 𝑋 → (𝑦 ∈ (𝐹 ∩ 𝐺) → (𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺)))))
31	30	imp 407	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (𝑦 ∈ (𝐹 ∩ 𝐺) → (𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺))))
32	31	rexlimdv 3138	. . 3 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ (𝐹 ∩ 𝐺)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺)))
33	32	ralrimiva 3131	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ (𝐹 ∩ 𝐺)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺)))
34		simpl 483	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
35		elinel1 4130	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 ∩ 𝐺) → 𝑥 ∈ 𝐹)
36	35, 17	anim12i 619	. . . . 5 ⊢ ((𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺)) → (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹))
37		filin 23837	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ∩ 𝑦) ∈ 𝐹)
38	37	3expb 1126	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ∈ 𝐹)
39	34, 36, 38	syl2an 602	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺))) → (𝑥 ∩ 𝑦) ∈ 𝐹)
40		simpr 485	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐺 ∈ (Fil‘𝑋))
41		elinel2 4131	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 ∩ 𝐺) → 𝑥 ∈ 𝐺)
42	41, 25	anim12i 619	. . . . 5 ⊢ ((𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺)) → (𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺))
43		filin 23837	. . . . . 6 ⊢ ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺) → (𝑥 ∩ 𝑦) ∈ 𝐺)
44	43	3expb 1126	. . . . 5 ⊢ ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺)) → (𝑥 ∩ 𝑦) ∈ 𝐺)
45	40, 42, 44	syl2an 602	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺))) → (𝑥 ∩ 𝑦) ∈ 𝐺)
46	39, 45	elind 4129	. . 3 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺))) → (𝑥 ∩ 𝑦) ∈ (𝐹 ∩ 𝐺))
47	46	ralrimivva 3182	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ (𝐹 ∩ 𝐺)∀𝑦 ∈ (𝐹 ∩ 𝐺)(𝑥 ∩ 𝑦) ∈ (𝐹 ∩ 𝐺))
48		isfil2 23839	. 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 1350	1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹 ∩ 𝐺) ∈ (Fil‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1092   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  𝒫 cpw 4529  ‘cfv 6485  Filcfil 23828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fv 6493  df-fbas 21344  df-fil 23829

This theorem is referenced by: (None)