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Theorem filin 22457
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 22451 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 fbasssin 22439 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))
31, 2syl3an1 1160 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))
4 inss1 4179 . . . . 5 (𝐴𝐵) ⊆ 𝐴
5 filelss 22455 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴𝑋)
64, 5sstrid 3953 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐴𝐵) ⊆ 𝑋)
7 filss 22456 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹 ∧ (𝐴𝐵) ⊆ 𝑋𝑥 ⊆ (𝐴𝐵))) → (𝐴𝐵) ∈ 𝐹)
873exp2 1351 . . . . . . 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 410 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐵) ⊆ 𝑋) → (𝑥𝐹 → (𝑥 ⊆ (𝐴𝐵) → (𝐴𝐵) ∈ 𝐹)))
1110rexlimdv 3269 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐵) ⊆ 𝑋) → (∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵) → (𝐴𝐵) ∈ 𝐹))
126, 11syldan 594 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵) → (𝐴𝐵) ∈ 𝐹))
13123adant3 1129 . 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 399  w3a 1084  wcel 2114  wrex 3131  cin 3907  wss 3908  cfv 6334  fBascfbas 20077  Filcfil 22448
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fv 6342  df-fbas 20086  df-fil 22449
This theorem is referenced by:  isfil2  22459  filfi  22462  filinn0  22463  infil  22466  filconn  22486  filuni  22488  trfil2  22490  trfilss  22492  ufprim  22512  filufint  22523  rnelfmlem  22555  rnelfm  22556  fmfnfmlem2  22558  fmfnfmlem3  22559  fmfnfmlem4  22560  fmfnfm  22561  txflf  22609  fclsrest  22627  metust  23163  filnetlem3  33802
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