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Theorem filin 21652
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 21646 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 fbasssin 21634 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))
31, 2syl3an1 1358 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))
4 inss1 3831 . . . . 5 (𝐴𝐵) ⊆ 𝐴
5 filelss 21650 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴𝑋)
64, 5syl5ss 3612 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐴𝐵) ⊆ 𝑋)
7 filss 21651 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹 ∧ (𝐴𝐵) ⊆ 𝑋𝑥 ⊆ (𝐴𝐵))) → (𝐴𝐵) ∈ 𝐹)
873exp2 1284 . . . . . . 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 445 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐵) ⊆ 𝑋) → (𝑥𝐹 → (𝑥 ⊆ (𝐴𝐵) → (𝐴𝐵) ∈ 𝐹)))
1110rexlimdv 3028 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐵) ⊆ 𝑋) → (∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵) → (𝐴𝐵) ∈ 𝐹))
126, 11syldan 487 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵) → (𝐴𝐵) ∈ 𝐹))
13123adant3 1080 . 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 384  w3a 1037  wcel 1989  wrex 2912  cin 3571  wss 3572  cfv 5886  fBascfbas 19728  Filcfil 21643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fv 5894  df-fbas 19737  df-fil 21644
This theorem is referenced by:  isfil2  21654  filfi  21657  filinn0  21658  infil  21661  filconn  21681  filuni  21683  trfil2  21685  trfilss  21687  ufprim  21707  filufint  21718  rnelfmlem  21750  rnelfm  21751  fmfnfmlem2  21753  fmfnfmlem3  21754  fmfnfmlem4  21755  fmfnfm  21756  txflf  21804  fclsrest  21822  metust  22357  filnetlem3  32359
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