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Theorem infil 23811
Description: The intersection of two filters is a filter. Use fiint 9350 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 4227 . . . 4 (𝐹𝐺) ⊆ 𝐹
2 filsspw 23799 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
32adantr 479 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
41, 3sstrid 3988 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹𝐺) ⊆ 𝒫 𝑋)
5 0nelfil 23797 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
65adantr 479 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
7 elinel1 4193 . . . 4 (∅ ∈ (𝐹𝐺) → ∅ ∈ 𝐹)
86, 7nsyl 140 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (𝐹𝐺))
9 filtop 23803 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
109adantr 479 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋𝐹)
11 filtop 23803 . . . . 5 (𝐺 ∈ (Fil‘𝑋) → 𝑋𝐺)
1211adantl 480 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋𝐺)
1310, 12elind 4192 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋 ∈ (𝐹𝐺))
144, 8, 133jca 1125 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ((𝐹𝐺) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ (𝐹𝐺) ∧ 𝑋 ∈ (𝐹𝐺)))
15 simpll 765 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝐹 ∈ (Fil‘𝑋))
16 simpr2 1192 . . . . . . . . 9 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦 ∈ (𝐹𝐺))
17 elinel1 4193 . . . . . . . . 9 (𝑦 ∈ (𝐹𝐺) → 𝑦𝐹)
1816, 17syl 17 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦𝐹)
19 simpr1 1191 . . . . . . . . 9 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥 ∈ 𝒫 𝑋)
2019elpwid 4613 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥𝑋)
21 simpr3 1193 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦𝑥)
22 filss 23801 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑦𝐹𝑥𝑋𝑦𝑥)) → 𝑥𝐹)
2315, 18, 20, 21, 22syl13anc 1369 . . . . . . 7 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥𝐹)
24 simplr 767 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝐺 ∈ (Fil‘𝑋))
25 elinel2 4194 . . . . . . . . 9 (𝑦 ∈ (𝐹𝐺) → 𝑦𝐺)
2616, 25syl 17 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦𝐺)
27 filss 23801 . . . . . . . 8 ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑦𝐺𝑥𝑋𝑦𝑥)) → 𝑥𝐺)
2824, 26, 20, 21, 27syl13anc 1369 . . . . . . 7 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥𝐺)
2923, 28elind 4192 . . . . . 6 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥 ∈ (𝐹𝐺))
30293exp2 1351 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝒫 𝑋 → (𝑦 ∈ (𝐹𝐺) → (𝑦𝑥𝑥 ∈ (𝐹𝐺)))))
3130imp 405 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (𝑦 ∈ (𝐹𝐺) → (𝑦𝑥𝑥 ∈ (𝐹𝐺))))
3231rexlimdv 3142 . . 3 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ (𝐹𝐺)𝑦𝑥𝑥 ∈ (𝐹𝐺)))
3332ralrimiva 3135 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ (𝐹𝐺)𝑦𝑥𝑥 ∈ (𝐹𝐺)))
34 simpl 481 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
35 elinel1 4193 . . . . . 6 (𝑥 ∈ (𝐹𝐺) → 𝑥𝐹)
3635, 17anim12i 611 . . . . 5 ((𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)) → (𝑥𝐹𝑦𝐹))
37 filin 23802 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹)
38373expb 1117 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹𝑦𝐹)) → (𝑥𝑦) ∈ 𝐹)
3934, 36, 38syl2an 594 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → (𝑥𝑦) ∈ 𝐹)
40 simpr 483 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐺 ∈ (Fil‘𝑋))
41 elinel2 4194 . . . . . 6 (𝑥 ∈ (𝐹𝐺) → 𝑥𝐺)
4241, 25anim12i 611 . . . . 5 ((𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)) → (𝑥𝐺𝑦𝐺))
43 filin 23802 . . . . . 6 ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥𝐺𝑦𝐺) → (𝑥𝑦) ∈ 𝐺)
44433expb 1117 . . . . 5 ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑥𝐺𝑦𝐺)) → (𝑥𝑦) ∈ 𝐺)
4540, 42, 44syl2an 594 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → (𝑥𝑦) ∈ 𝐺)
4639, 45elind 4192 . . 3 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → (𝑥𝑦) ∈ (𝐹𝐺))
4746ralrimivva 3190 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ (𝐹𝐺)∀𝑦 ∈ (𝐹𝐺)(𝑥𝑦) ∈ (𝐹𝐺))
48 isfil2 23804 . 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 1340 1 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹𝐺) ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3a 1084  wcel 2098  wral 3050  wrex 3059  cin 3943  wss 3944  c0 4322  𝒫 cpw 4604  cfv 6549  Filcfil 23793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fv 6557  df-fbas 21293  df-fil 23794
This theorem is referenced by: (None)
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