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Theorem infil 21607
Description: The intersection of two filters is a filter. Use fiint 8197 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 3817 . . . 4 (𝐹𝐺) ⊆ 𝐹
2 filsspw 21595 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
32adantr 481 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
41, 3syl5ss 3599 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹𝐺) ⊆ 𝒫 𝑋)
5 0nelfil 21593 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
65adantr 481 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
71sseli 3584 . . . 4 (∅ ∈ (𝐹𝐺) → ∅ ∈ 𝐹)
86, 7nsyl 135 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (𝐹𝐺))
9 filtop 21599 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
109adantr 481 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋𝐹)
11 filtop 21599 . . . . 5 (𝐺 ∈ (Fil‘𝑋) → 𝑋𝐺)
1211adantl 482 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋𝐺)
1310, 12elind 3782 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋 ∈ (𝐹𝐺))
144, 8, 133jca 1240 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ((𝐹𝐺) ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ (𝐹𝐺) ∧ 𝑋 ∈ (𝐹𝐺)))
15 simpll 789 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝐹 ∈ (Fil‘𝑋))
16 simpr2 1066 . . . . . . . . 9 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦 ∈ (𝐹𝐺))
171sseli 3584 . . . . . . . . 9 (𝑦 ∈ (𝐹𝐺) → 𝑦𝐹)
1816, 17syl 17 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦𝐹)
19 simpr1 1065 . . . . . . . . 9 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥 ∈ 𝒫 𝑋)
2019elpwid 4148 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥𝑋)
21 simpr3 1067 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦𝑥)
22 filss 21597 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑦𝐹𝑥𝑋𝑦𝑥)) → 𝑥𝐹)
2315, 18, 20, 21, 22syl13anc 1325 . . . . . . 7 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥𝐹)
24 simplr 791 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝐺 ∈ (Fil‘𝑋))
25 inss2 3818 . . . . . . . . . 10 (𝐹𝐺) ⊆ 𝐺
2625sseli 3584 . . . . . . . . 9 (𝑦 ∈ (𝐹𝐺) → 𝑦𝐺)
2716, 26syl 17 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦𝐺)
28 filss 21597 . . . . . . . 8 ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑦𝐺𝑥𝑋𝑦𝑥)) → 𝑥𝐺)
2924, 27, 20, 21, 28syl13anc 1325 . . . . . . 7 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥𝐺)
3023, 29elind 3782 . . . . . 6 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑥 ∈ (𝐹𝐺))
31303exp2 1282 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝑥 ∈ 𝒫 𝑋 → (𝑦 ∈ (𝐹𝐺) → (𝑦𝑥𝑥 ∈ (𝐹𝐺)))))
3231imp 445 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (𝑦 ∈ (𝐹𝐺) → (𝑦𝑥𝑥 ∈ (𝐹𝐺))))
3332rexlimdv 3025 . . 3 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ (𝐹𝐺)𝑦𝑥𝑥 ∈ (𝐹𝐺)))
3433ralrimiva 2962 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ (𝐹𝐺)𝑦𝑥𝑥 ∈ (𝐹𝐺)))
35 simpl 473 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
361sseli 3584 . . . . . 6 (𝑥 ∈ (𝐹𝐺) → 𝑥𝐹)
3736, 17anim12i 589 . . . . 5 ((𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)) → (𝑥𝐹𝑦𝐹))
38 filin 21598 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹)
39383expb 1263 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹𝑦𝐹)) → (𝑥𝑦) ∈ 𝐹)
4035, 37, 39syl2an 494 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → (𝑥𝑦) ∈ 𝐹)
41 simpr 477 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐺 ∈ (Fil‘𝑋))
4225sseli 3584 . . . . . 6 (𝑥 ∈ (𝐹𝐺) → 𝑥𝐺)
4342, 26anim12i 589 . . . . 5 ((𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)) → (𝑥𝐺𝑦𝐺))
44 filin 21598 . . . . . 6 ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥𝐺𝑦𝐺) → (𝑥𝑦) ∈ 𝐺)
45443expb 1263 . . . . 5 ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑥𝐺𝑦𝐺)) → (𝑥𝑦) ∈ 𝐺)
4641, 43, 45syl2an 494 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → (𝑥𝑦) ∈ 𝐺)
4740, 46elind 3782 . . 3 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → (𝑥𝑦) ∈ (𝐹𝐺))
4847ralrimivva 2967 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ (𝐹𝐺)∀𝑦 ∈ (𝐹𝐺)(𝑥𝑦) ∈ (𝐹𝐺))
49 isfil2 21600 . 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 1244 1 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹𝐺) ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036  wcel 1987  wral 2908  wrex 2909  cin 3559  wss 3560  c0 3897  𝒫 cpw 4136  cfv 5857  Filcfil 21589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fv 5865  df-fbas 19683  df-fil 21590
This theorem is referenced by: (None)
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