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Theorem infil 22472
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 4158 . . . 4 (𝐹𝐺) ⊆ 𝐹
2 filsspw 22460 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
32adantr 484 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ⊆ 𝒫 𝑋)
41, 3sstrid 3929 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹𝐺) ⊆ 𝒫 𝑋)
5 0nelfil 22458 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
65adantr 484 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ 𝐹)
7 elinel1 4125 . . . 4 (∅ ∈ (𝐹𝐺) → ∅ ∈ 𝐹)
86, 7nsyl 142 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ¬ ∅ ∈ (𝐹𝐺))
9 filtop 22464 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
109adantr 484 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋𝐹)
11 filtop 22464 . . . . 5 (𝐺 ∈ (Fil‘𝑋) → 𝑋𝐺)
1211adantl 485 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝑋𝐺)
1310, 12elind 4124 . . 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 766 . . . . . . . 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 4125 . . . . . . . . 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 4511 . . . . . . . 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 22462 . . . . . . . 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 768 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝐺 ∈ (Fil‘𝑋))
25 elinel2 4126 . . . . . . . . 9 (𝑦 ∈ (𝐹𝐺) → 𝑦𝐺)
2616, 25syl 17 . . . . . . . 8 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ 𝒫 𝑋𝑦 ∈ (𝐹𝐺) ∧ 𝑦𝑥)) → 𝑦𝐺)
27 filss 22462 . . . . . . . 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 4124 . . . . . 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 410 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (𝑦 ∈ (𝐹𝐺) → (𝑦𝑥𝑥 ∈ (𝐹𝐺))))
3231rexlimdv 3245 . . 3 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝒫 𝑋) → (∃𝑦 ∈ (𝐹𝐺)𝑦𝑥𝑥 ∈ (𝐹𝐺)))
3332ralrimiva 3152 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ (𝐹𝐺)𝑦𝑥𝑥 ∈ (𝐹𝐺)))
34 simpl 486 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐹 ∈ (Fil‘𝑋))
35 elinel1 4125 . . . . . 6 (𝑥 ∈ (𝐹𝐺) → 𝑥𝐹)
3635, 17anim12i 615 . . . . 5 ((𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)) → (𝑥𝐹𝑦𝐹))
37 filin 22463 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹)
38373expb 1117 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹𝑦𝐹)) → (𝑥𝑦) ∈ 𝐹)
3934, 36, 38syl2an 598 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → (𝑥𝑦) ∈ 𝐹)
40 simpr 488 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → 𝐺 ∈ (Fil‘𝑋))
41 elinel2 4126 . . . . . 6 (𝑥 ∈ (𝐹𝐺) → 𝑥𝐺)
4241, 25anim12i 615 . . . . 5 ((𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺)) → (𝑥𝐺𝑦𝐺))
43 filin 22463 . . . . . 6 ((𝐺 ∈ (Fil‘𝑋) ∧ 𝑥𝐺𝑦𝐺) → (𝑥𝑦) ∈ 𝐺)
44433expb 1117 . . . . 5 ((𝐺 ∈ (Fil‘𝑋) ∧ (𝑥𝐺𝑦𝐺)) → (𝑥𝑦) ∈ 𝐺)
4540, 42, 44syl2an 598 . . . 4 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → (𝑥𝑦) ∈ 𝐺)
4639, 45elind 4124 . . 3 (((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) ∧ (𝑥 ∈ (𝐹𝐺) ∧ 𝑦 ∈ (𝐹𝐺))) → (𝑥𝑦) ∈ (𝐹𝐺))
4746ralrimivva 3159 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → ∀𝑥 ∈ (𝐹𝐺)∀𝑦 ∈ (𝐹𝐺)(𝑥𝑦) ∈ (𝐹𝐺))
48 isfil2 22465 . 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 399  w3a 1084  wcel 2112  wral 3109  wrex 3110  cin 3883  wss 3884  c0 4246  𝒫 cpw 4500  cfv 6328  Filcfil 22454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336  df-fbas 20092  df-fil 22455
This theorem is referenced by: (None)
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