MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  infil Structured version   Visualization version   GIF version

Theorem infil 22399
Description: The intersection of two filters is a filter. Use fiint 8783 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 4202 . . . 4 (𝐹𝐺) ⊆ 𝐹
2 filsspw 22387 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
32adantr 481 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
41, 3sstrid 3975 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹𝐺) ⊆ 𝒫 𝑋)
5 0nelfil 22385 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
65adantr 481 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
7 elinel1 4169 . . . 4 (∅ ∈ (𝐹𝐺) → ∅ ∈ 𝐹)
86, 7nsyl 142 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (𝐹𝐺))
9 filtop 22391 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
109adantr 481 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋𝐹)
11 filtop 22391 . . . . 5 (𝐺 ∈ (Fil‘𝑋) → 𝑋𝐺)
1211adantl 482 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋𝐺)
1310, 12elind 4168 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋 ∈ (𝐹𝐺))
144, 8, 133jca 1120 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ((𝐹𝐺) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ (𝐹𝐺) ∧ 𝑋 ∈ (𝐹𝐺)))
15 simpll 763 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝐹 ∈ (Fil‘𝑋))
16 simpr2 1187 . . . . . . . . 9 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦 ∈ (𝐹𝐺))
17 elinel1 4169 . . . . . . . . 9 (𝑦 ∈ (𝐹𝐺) → 𝑦𝐹)
1816, 17syl 17 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦𝐹)
19 simpr1 1186 . . . . . . . . 9 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥 ∈ 𝒫 𝑋)
2019elpwid 4549 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥𝑋)
21 simpr3 1188 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦𝑥)
22 filss 22389 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑦𝐹𝑥𝑋𝑦𝑥)) → 𝑥𝐹)
2315, 18, 20, 21, 22syl13anc 1364 . . . . . . 7 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥𝐹)
24 simplr 765 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝐺 ∈ (Fil‘𝑋))
25 elinel2 4170 . . . . . . . . 9 (𝑦 ∈ (𝐹𝐺) → 𝑦𝐺)
2616, 25syl 17 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦𝐺)
27 filss 22389 . . . . . . . 8 ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑦𝐺𝑥𝑋𝑦𝑥)) → 𝑥𝐺)
2824, 26, 20, 21, 27syl13anc 1364 . . . . . . 7 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥𝐺)
2923, 28elind 4168 . . . . . 6 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥 ∈ (𝐹𝐺))
30293exp2 1346 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝒫 𝑋 → (𝑦 ∈ (𝐹𝐺) → (𝑦𝑥𝑥 ∈ (𝐹𝐺)))))
3130imp 407 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (𝑦 ∈ (𝐹𝐺) → (𝑦𝑥𝑥 ∈ (𝐹𝐺))))
3231rexlimdv 3280 . . 3 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ (𝐹𝐺)𝑦𝑥𝑥 ∈ (𝐹𝐺)))
3332ralrimiva 3179 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ (𝐹𝐺)𝑦𝑥𝑥 ∈ (𝐹𝐺)))
34 simpl 483 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
35 elinel1 4169 . . . . . 6 (𝑥 ∈ (𝐹𝐺) → 𝑥𝐹)
3635, 17anim12i 612 . . . . 5 ((𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)) → (𝑥𝐹𝑦𝐹))
37 filin 22390 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹)
38373expb 1112 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹𝑦𝐹)) → (𝑥𝑦) ∈ 𝐹)
3934, 36, 38syl2an 595 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → (𝑥𝑦) ∈ 𝐹)
40 simpr 485 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐺 ∈ (Fil‘𝑋))
41 elinel2 4170 . . . . . 6 (𝑥 ∈ (𝐹𝐺) → 𝑥𝐺)
4241, 25anim12i 612 . . . . 5 ((𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)) → (𝑥𝐺𝑦𝐺))
43 filin 22390 . . . . . 6 ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥𝐺𝑦𝐺) → (𝑥𝑦) ∈ 𝐺)
44433expb 1112 . . . . 5 ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑥𝐺𝑦𝐺)) → (𝑥𝑦) ∈ 𝐺)
4540, 42, 44syl2an 595 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → (𝑥𝑦) ∈ 𝐺)
4639, 45elind 4168 . . 3 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → (𝑥𝑦) ∈ (𝐹𝐺))
4746ralrimivva 3188 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ (𝐹𝐺)∀𝑦 ∈ (𝐹𝐺)(𝑥𝑦) ∈ (𝐹𝐺))
48 isfil2 22392 . 2 ((𝐹𝐺) ∈ (Fil‘𝑋) ↔ (((𝐹𝐺) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ (𝐹𝐺) ∧ 𝑋 ∈ (𝐹𝐺)) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ (𝐹𝐺)𝑦𝑥𝑥 ∈ (𝐹𝐺)) ∧ ∀𝑥 ∈ (𝐹𝐺)∀𝑦 ∈ (𝐹𝐺)(𝑥𝑦) ∈ (𝐹𝐺)))
4914, 33, 47, 48syl3anbrc 1335 1 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹𝐺) ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079  wcel 2105  wral 3135  wrex 3136  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535  cfv 6348  Filcfil 22381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-fbas 20470  df-fil 22382
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator