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Theorem filfi 22467
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 22462 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹)
213expib 1119 . . 3 (𝐹 ∈ (Fil‘𝑋) → ((𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹))
32ralrimivv 3185 . 2 (𝐹 ∈ (Fil‘𝑋) → ∀𝑥𝐹𝑦𝐹 (𝑥𝑦) ∈ 𝐹)
4 inficl 8886 . 2 (𝐹 ∈ (Fil‘𝑋) → (∀𝑥𝐹𝑦𝐹 (𝑥𝑦) ∈ 𝐹 ↔ (fi‘𝐹) = 𝐹))
53, 4mpbid 235 1 (𝐹 ∈ (Fil‘𝑋) → (fi‘𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wral 3133  cin 3918  cfv 6343  ficfi 8871  Filcfil 22453
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-en 8506  df-fin 8509  df-fi 8872  df-fbas 20542  df-fil 22454
This theorem is referenced by:  filintn0  22469  fclscmpi  22637  alexsublem  22652  iscmet3  23900
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