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Theorem filfi 21573
 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 21568 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹)
213expib 1265 . . 3 (𝐹 ∈ (Fil‘𝑋) → ((𝑥𝐹𝑦𝐹) → (𝑥𝑦) ∈ 𝐹))
32ralrimivv 2964 . 2 (𝐹 ∈ (Fil‘𝑋) → ∀𝑥𝐹𝑦𝐹 (𝑥𝑦) ∈ 𝐹)
4 inficl 8275 . 2 (𝐹 ∈ (Fil‘𝑋) → (∀𝑥𝐹𝑦𝐹 (𝑥𝑦) ∈ 𝐹 ↔ (fi‘𝐹) = 𝐹))
53, 4mpbid 222 1 (𝐹 ∈ (Fil‘𝑋) → (fi‘𝐹) = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  ∀wral 2907   ∩ cin 3554  ‘cfv 5847  ficfi 8260  Filcfil 21559 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-fin 7903  df-fi 8261  df-fbas 19662  df-fil 21560 This theorem is referenced by:  filintn0  21575  fclscmpi  21743  alexsublem  21758  iscmet3  22999
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