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Theorem filfi 22075
 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 22070 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹)
213expib 1113 . . 3 (𝐹 ∈ (Fil‘𝑋) → ((𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹))
32ralrimivv 3152 . 2 (𝐹 ∈ (Fil‘𝑋) → ∀𝑥𝐹𝑦𝐹 (𝑥𝑦) ∈ 𝐹)
4 inficl 8621 . 2 (𝐹 ∈ (Fil‘𝑋) → (∀𝑥𝐹𝑦𝐹 (𝑥𝑦) ∈ 𝐹 ↔ (fi‘𝐹) = 𝐹))
53, 4mpbid 224 1 (𝐹 ∈ (Fil‘𝑋) → (fi‘𝐹) = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2107  ∀wral 3090   ∩ cin 3791  ‘cfv 6137  ficfi 8606  Filcfil 22061 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-er 8028  df-en 8244  df-fin 8247  df-fi 8607  df-fbas 20143  df-fil 22062 This theorem is referenced by:  filintn0  22077  fclscmpi  22245  alexsublem  22260  iscmet3  23503
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