Theorem filin 23739

Description: A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)

Assertion

Ref	Expression
filin	⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (𝐴 ∩ 𝐵) ∈ 𝐹)

Proof of Theorem filin

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		filfbas 23733	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2		fbasssin 23721	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵))
3	1, 2	syl3an1 1163	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵))
4		inss1 4188	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
5		filelss 23737	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋)
6	4, 5	sstrid 3947	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐴 ∩ 𝐵) ⊆ 𝑋)
7		filss 23738	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ (𝐴 ∩ 𝐵) ⊆ 𝑋 ∧ 𝑥 ⊆ (𝐴 ∩ 𝐵))) → (𝐴 ∩ 𝐵) ∈ 𝐹)
8	7	3exp2 1355	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐹 → ((𝐴 ∩ 𝐵) ⊆ 𝑋 → (𝑥 ⊆ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐹))))
9	8	com23 86	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝐴 ∩ 𝐵) ⊆ 𝑋 → (𝑥 ∈ 𝐹 → (𝑥 ⊆ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐹))))
10	9	imp 406	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∩ 𝐵) ⊆ 𝑋) → (𝑥 ∈ 𝐹 → (𝑥 ⊆ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐹)))
11	10	rexlimdv 3128	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∩ 𝐵) ⊆ 𝑋) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐹))
12	6, 11	syldan 591	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐹))
13	12	3adant3 1132	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐹))
14	3, 13	mpd 15	1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (𝐴 ∩ 𝐵) ∈ 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109  ∃wrex 3053   ∩ cin 3902   ⊆ wss 3903  ‘cfv 6482  fBascfbas 21249  Filcfil 23730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fv 6490  df-fbas 21258  df-fil 23731

This theorem is referenced by:  isfil2  23741  filfi  23744  filinn0  23745  infil  23748  filconn  23768  filuni  23770  trfil2  23772  trfilss  23774  ufprim  23794  filufint  23805  rnelfmlem  23837  rnelfm  23838  fmfnfmlem2  23840  fmfnfmlem3  23841  fmfnfmlem4  23842  fmfnfm  23843  txflf  23891  fclsrest  23909  metust  24444  filnetlem3  36364