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Theorem filfi 23868
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 23863 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹)
213expib 1122 . . 3 (𝐹 ∈ (Fil‘𝑋) → ((𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹))
32ralrimivv 3199 . 2 (𝐹 ∈ (Fil‘𝑋) → ∀𝑥𝐹𝑦𝐹 (𝑥𝑦) ∈ 𝐹)
4 inficl 9466 . 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 1539  wcel 2107  wral 3060  cin 3949  cfv 6560  ficfi 9451  Filcfil 23854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-om 7889  df-1o 8507  df-2o 8508  df-en 8987  df-fin 8990  df-fi 9452  df-fbas 21362  df-fil 23855
This theorem is referenced by:  filintn0  23870  fclscmpi  24038  alexsublem  24053  iscmet3  25328
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