Theorem infil 24182

Description: The intersection of two filters is a filter. Use fiint 9318 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 4182	. . . 4 ⊢ (𝐹 ∩ 𝐺) ⊆ 𝐹
2		filsspw 24170	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
3	2	adantr 486	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
4	1, 3	sstrid 3942	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹 ∩ 𝐺) ⊆ 𝒫 𝑋)
5		0nelfil 24168	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
6	5	adantr 486	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
7		elinel1 4147	. . . 4 ⊢ (∅ ∈ (𝐹 ∩ 𝐺) → ∅ ∈ 𝐹)
8	6, 7	nsyl 141	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (𝐹 ∩ 𝐺))
9		filtop 24174	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
10	9	adantr 486	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋 ∈ 𝐹)
11		filtop 24174	. . . . 5 ⊢ (𝐺 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐺)
12	11	adantl 487	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋 ∈ 𝐺)
13	10, 12	elind 4146	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋 ∈ (𝐹 ∩ 𝐺))
14	4, 8, 13	3jca 1146	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ((𝐹 ∩ 𝐺) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ (𝐹 ∩ 𝐺) ∧ 𝑋 ∈ (𝐹 ∩ 𝐺)))
15		simpll 779	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝐹 ∈ (Fil‘𝑋))
16		simpr2 1214	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ∈ (𝐹 ∩ 𝐺))
17		elinel1 4147	. . . . . . . . 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 1213	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝒫 𝑋)
20	19	elpwid 4566	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ⊆ 𝑋)
21		simpr3 1215	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑦 ⊆ 𝑥)
22		filss 24172	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐹 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐹)
23	15, 18, 20, 21, 22	syl13anc 1399	. . . . . . 7 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐹)
24		simplr 781	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝐺 ∈ (Fil‘𝑋))
25		elinel2 4148	. . . . . . . . 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 24172	. . . . . . . 8 ⊢ ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐺 ∧ 𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐺)
28	24, 26, 20, 21, 27	syl13anc 1399	. . . . . . 7 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ 𝐺)
29	23, 28	elind 4146	. . . . . 6 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋 ∧ 𝑦 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ⊆ 𝑥)) → 𝑥 ∈ (𝐹 ∩ 𝐺))
30	29	3exp2 1373	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝒫 𝑋 → (𝑦 ∈ (𝐹 ∩ 𝐺) → (𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺)))))
31	30	imp 412	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (𝑦 ∈ (𝐹 ∩ 𝐺) → (𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺))))
32	31	rexlimdv 3162	. . 3 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ (𝐹 ∩ 𝐺)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺)))
33	32	ralrimiva 3155	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ (𝐹 ∩ 𝐺)𝑦 ⊆ 𝑥 → 𝑥 ∈ (𝐹 ∩ 𝐺)))
34		simpl 488	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
35		elinel1 4147	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 ∩ 𝐺) → 𝑥 ∈ 𝐹)
36	35, 17	anim12i 625	. . . . 5 ⊢ ((𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺)) → (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹))
37		filin 24173	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ∩ 𝑦) ∈ 𝐹)
38	37	3expb 1138	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 ∩ 𝑦) ∈ 𝐹)
39	34, 36, 38	syl2an 608	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺))) → (𝑥 ∩ 𝑦) ∈ 𝐹)
40		simpr 490	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐺 ∈ (Fil‘𝑋))
41		elinel2 4148	. . . . . 6 ⊢ (𝑥 ∈ (𝐹 ∩ 𝐺) → 𝑥 ∈ 𝐺)
42	41, 25	anim12i 625	. . . . 5 ⊢ ((𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺)) → (𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺))
43		filin 24173	. . . . . 6 ⊢ ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺) → (𝑥 ∩ 𝑦) ∈ 𝐺)
44	43	3expb 1138	. . . . 5 ⊢ ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐺 ∧ 𝑦 ∈ 𝐺)) → (𝑥 ∩ 𝑦) ∈ 𝐺)
45	40, 42, 44	syl2an 608	. . . 4 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺))) → (𝑥 ∩ 𝑦) ∈ 𝐺)
46	39, 45	elind 4146	. . 3 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹 ∩ 𝐺) ∧ 𝑦 ∈ (𝐹 ∩ 𝐺))) → (𝑥 ∩ 𝑦) ∈ (𝐹 ∩ 𝐺))
47	46	ralrimivva 3206	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ (𝐹 ∩ 𝐺)∀𝑦 ∈ (𝐹 ∩ 𝐺)(𝑥 ∩ 𝑦) ∈ (𝐹 ∩ 𝐺))
48		isfil2 24175	. 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 1362	1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹 ∩ 𝐺) ∈ (Fil‘𝑋))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  ‘cfv 6538  Filcfil 24164

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fv 6546  df-fbas 21675  df-fil 24165

This theorem is used by: (None)