MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  filin Structured version   Visualization version   GIF version

Theorem filin 23358
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 23352 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 fbasssin 23340 . . 3 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))
31, 2syl3an1 1164 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))
4 inss1 4229 . . . . 5 (𝐴𝐵) ⊆ 𝐴
5 filelss 23356 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴𝑋)
64, 5sstrid 3994 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐴𝐵) ⊆ 𝑋)
7 filss 23357 . . . . . . . 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 3948  wss 3949  cfv 6544  fBascfbas 20932  Filcfil 23349
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-fbas 20941  df-fil 23350
This theorem is referenced by:  isfil2  23360  filfi  23363  filinn0  23364  infil  23367  filconn  23387  filuni  23389  trfil2  23391  trfilss  23393  ufprim  23413  filufint  23424  rnelfmlem  23456  rnelfm  23457  fmfnfmlem2  23459  fmfnfmlem3  23460  fmfnfmlem4  23461  fmfnfm  23462  txflf  23510  fclsrest  23528  metust  24067  filnetlem3  35265
  Copyright terms: Public domain W3C validator