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Theorem filin 22077
Description: A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
filin ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (𝐴𝐵) ∈ 𝐹)

Proof of Theorem filin
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 filfbas 22071 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 fbasssin 22059 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))
31, 2syl3an1 1163 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))
4 inss1 4053 . . . . 5 (𝐴𝐵) ⊆ 𝐴
5 filelss 22075 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴𝑋)
64, 5syl5ss 3832 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐴𝐵) ⊆ 𝑋)
7 filss 22076 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹 ∧ (𝐴𝐵) ⊆ 𝑋𝑥 ⊆ (𝐴𝐵))) → (𝐴𝐵) ∈ 𝐹)
873exp2 1416 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹 → ((𝐴𝐵) ⊆ 𝑋 → (𝑥 ⊆ (𝐴𝐵) → (𝐴𝐵) ∈ 𝐹))))
98com23 86 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → ((𝐴𝐵) ⊆ 𝑋 → (𝑥𝐹 → (𝑥 ⊆ (𝐴𝐵) → (𝐴𝐵) ∈ 𝐹))))
109imp 397 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐵) ⊆ 𝑋) → (𝑥𝐹 → (𝑥 ⊆ (𝐴𝐵) → (𝐴𝐵) ∈ 𝐹)))
1110rexlimdv 3212 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐵) ⊆ 𝑋) → (∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵) → (𝐴𝐵) ∈ 𝐹))
126, 11syldan 585 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵) → (𝐴𝐵) ∈ 𝐹))
13123adant3 1123 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵) → (𝐴𝐵) ∈ 𝐹))
143, 13mpd 15 1 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (𝐴𝐵) ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071  wcel 2107  wrex 3091  cin 3791  wss 3792  cfv 6137  fBascfbas 20141  Filcfil 22068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fv 6145  df-fbas 20150  df-fil 22069
This theorem is referenced by:  isfil2  22079  filfi  22082  filinn0  22083  infil  22086  filconn  22106  filuni  22108  trfil2  22110  trfilss  22112  ufprim  22132  filufint  22143  rnelfmlem  22175  rnelfm  22176  fmfnfmlem2  22178  fmfnfmlem3  22179  fmfnfmlem4  22180  fmfnfm  22181  txflf  22229  fclsrest  22247  metust  22782  filnetlem3  32971
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