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

Theorem filfi 23883
Description: A filter is closed under taking intersections. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
filfi (𝐹 ∈ (Fil‘𝑋) → (fi‘𝐹) = 𝐹)

Proof of Theorem filfi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 filin 23878 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹)
213expib 1121 . . 3 (𝐹 ∈ (Fil‘𝑋) → ((𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹))
32ralrimivv 3198 . 2 (𝐹 ∈ (Fil‘𝑋) → ∀𝑥𝐹𝑦𝐹 (𝑥𝑦) ∈ 𝐹)
4 inficl 9463 . 2 (𝐹 ∈ (Fil‘𝑋) → (∀𝑥𝐹𝑦𝐹 (𝑥𝑦) ∈ 𝐹 ↔ (fi‘𝐹) = 𝐹))
53, 4mpbid 232 1 (𝐹 ∈ (Fil‘𝑋) → (fi‘𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059  cin 3962  cfv 6563  ficfi 9448  Filcfil 23869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-2o 8506  df-en 8985  df-fin 8988  df-fi 9449  df-fbas 21379  df-fil 23870
This theorem is referenced by:  filintn0  23885  fclscmpi  24053  alexsublem  24068  iscmet3  25341
  Copyright terms: Public domain W3C validator