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Theorem filin 23327
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 23321 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 fbasssin 23309 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))
31, 2syl3an1 1164 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))
4 inss1 4226 . . . . 5 (𝐴𝐵) ⊆ 𝐴
5 filelss 23325 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴𝑋)
64, 5sstrid 3991 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐴𝐵) ⊆ 𝑋)
7 filss 23326 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹 ∧ (𝐴𝐵) ⊆ 𝑋𝑥 ⊆ (𝐴𝐵))) → (𝐴𝐵) ∈ 𝐹)
873exp2 1355 . . . . . . 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 408 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐵) ⊆ 𝑋) → (𝑥𝐹 → (𝑥 ⊆ (𝐴𝐵) → (𝐴𝐵) ∈ 𝐹)))
1110rexlimdv 3154 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐵) ⊆ 𝑋) → (∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵) → (𝐴𝐵) ∈ 𝐹))
126, 11syldan 592 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵) → (𝐴𝐵) ∈ 𝐹))
13123adant3 1133 . 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 397  w3a 1088  wcel 2107  wrex 3071  cin 3945  wss 3946  cfv 6535  fBascfbas 20906  Filcfil 23318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fv 6543  df-fbas 20915  df-fil 23319
This theorem is referenced by:  isfil2  23329  filfi  23332  filinn0  23333  infil  23336  filconn  23356  filuni  23358  trfil2  23360  trfilss  23362  ufprim  23382  filufint  23393  rnelfmlem  23425  rnelfm  23426  fmfnfmlem2  23428  fmfnfmlem3  23429  fmfnfmlem4  23430  fmfnfm  23431  txflf  23479  fclsrest  23497  metust  24036  filnetlem3  35170
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