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Theorem infil 21888
Description: The intersection of two filters is a filter. Use fiint 8404 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)) → (𝐹𝐺) ∈ (Fil‘𝑋))

Proof of Theorem infil
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 3976 . . . 4 (𝐹𝐺) ⊆ 𝐹
2 filsspw 21876 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
32adantr 472 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
41, 3syl5ss 3755 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹𝐺) ⊆ 𝒫 𝑋)
5 0nelfil 21874 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
65adantr 472 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
71sseli 3740 . . . 4 (∅ ∈ (𝐹𝐺) → ∅ ∈ 𝐹)
86, 7nsyl 135 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (𝐹𝐺))
9 filtop 21880 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
109adantr 472 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋𝐹)
11 filtop 21880 . . . . 5 (𝐺 ∈ (Fil‘𝑋) → 𝑋𝐺)
1211adantl 473 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋𝐺)
1310, 12elind 3941 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋 ∈ (𝐹𝐺))
144, 8, 133jca 1123 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ((𝐹𝐺) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ (𝐹𝐺) ∧ 𝑋 ∈ (𝐹𝐺)))
15 simpll 807 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝐹 ∈ (Fil‘𝑋))
16 simpr2 1236 . . . . . . . . 9 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦 ∈ (𝐹𝐺))
171sseli 3740 . . . . . . . . 9 (𝑦 ∈ (𝐹𝐺) → 𝑦𝐹)
1816, 17syl 17 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦𝐹)
19 simpr1 1234 . . . . . . . . 9 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥 ∈ 𝒫 𝑋)
2019elpwid 4314 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥𝑋)
21 simpr3 1238 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦𝑥)
22 filss 21878 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑦𝐹𝑥𝑋𝑦𝑥)) → 𝑥𝐹)
2315, 18, 20, 21, 22syl13anc 1479 . . . . . . 7 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥𝐹)
24 simplr 809 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝐺 ∈ (Fil‘𝑋))
25 inss2 3977 . . . . . . . . . 10 (𝐹𝐺) ⊆ 𝐺
2625sseli 3740 . . . . . . . . 9 (𝑦 ∈ (𝐹𝐺) → 𝑦𝐺)
2716, 26syl 17 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦𝐺)
28 filss 21878 . . . . . . . 8 ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑦𝐺𝑥𝑋𝑦𝑥)) → 𝑥𝐺)
2924, 27, 20, 21, 28syl13anc 1479 . . . . . . 7 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥𝐺)
3023, 29elind 3941 . . . . . 6 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥 ∈ (𝐹𝐺))
31303exp2 1448 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝒫 𝑋 → (𝑦 ∈ (𝐹𝐺) → (𝑦𝑥𝑥 ∈ (𝐹𝐺)))))
3231imp 444 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (𝑦 ∈ (𝐹𝐺) → (𝑦𝑥𝑥 ∈ (𝐹𝐺))))
3332rexlimdv 3168 . . 3 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ (𝐹𝐺)𝑦𝑥𝑥 ∈ (𝐹𝐺)))
3433ralrimiva 3104 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ (𝐹𝐺)𝑦𝑥𝑥 ∈ (𝐹𝐺)))
35 simpl 474 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
361sseli 3740 . . . . . 6 (𝑥 ∈ (𝐹𝐺) → 𝑥𝐹)
3736, 17anim12i 591 . . . . 5 ((𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)) → (𝑥𝐹𝑦𝐹))
38 filin 21879 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹)
39383expb 1114 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹𝑦𝐹)) → (𝑥𝑦) ∈ 𝐹)
4035, 37, 39syl2an 495 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → (𝑥𝑦) ∈ 𝐹)
41 simpr 479 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐺 ∈ (Fil‘𝑋))
4225sseli 3740 . . . . . 6 (𝑥 ∈ (𝐹𝐺) → 𝑥𝐺)
4342, 26anim12i 591 . . . . 5 ((𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)) → (𝑥𝐺𝑦𝐺))
44 filin 21879 . . . . . 6 ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥𝐺𝑦𝐺) → (𝑥𝑦) ∈ 𝐺)
45443expb 1114 . . . . 5 ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑥𝐺𝑦𝐺)) → (𝑥𝑦) ∈ 𝐺)
4641, 43, 45syl2an 495 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → (𝑥𝑦) ∈ 𝐺)
4740, 46elind 3941 . . 3 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → (𝑥𝑦) ∈ (𝐹𝐺))
4847ralrimivva 3109 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ (𝐹𝐺)∀𝑦 ∈ (𝐹𝐺)(𝑥𝑦) ∈ (𝐹𝐺))
49 isfil2 21881 . 2 ((𝐹𝐺) ∈ (Fil‘𝑋) ↔ (((𝐹𝐺) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ (𝐹𝐺) ∧ 𝑋 ∈ (𝐹𝐺)) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ (𝐹𝐺)𝑦𝑥𝑥 ∈ (𝐹𝐺)) ∧ ∀𝑥 ∈ (𝐹𝐺)∀𝑦 ∈ (𝐹𝐺)(𝑥𝑦) ∈ (𝐹𝐺)))
5014, 34, 48, 49syl3anbrc 1429 1 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹𝐺) ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1072  wcel 2139  wral 3050  wrex 3051  cin 3714  wss 3715  c0 4058  𝒫 cpw 4302  cfv 6049  Filcfil 21870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-fbas 19965  df-fil 21871
This theorem is referenced by: (None)
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