Theorem filin 23844

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 23838	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2		fbasssin 23826	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵))
3	1, 2	syl3an1 1160	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵))
4		inss1 4228	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
5		filelss 23842	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋)
6	4, 5	sstrid 3991	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐴 ∩ 𝐵) ⊆ 𝑋)
7		filss 23843	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ (𝐴 ∩ 𝐵) ⊆ 𝑋 ∧ 𝑥 ⊆ (𝐴 ∩ 𝐵))) → (𝐴 ∩ 𝐵) ∈ 𝐹)
8	7	3exp2 1351	. . . . . . 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 405	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∩ 𝐵) ⊆ 𝑋) → (𝑥 ∈ 𝐹 → (𝑥 ⊆ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐹)))
11	10	rexlimdv 3143	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∩ 𝐵) ⊆ 𝑋) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐹))
12	6, 11	syldan 589	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐹))
13	12	3adant3 1129	. 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 394   ∧ w3a 1084   ∈ wcel 2099  ∃wrex 3060   ∩ cin 3946   ⊆ wss 3947  ‘cfv 6544  fBascfbas 21325  Filcfil 23835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6496  df-fun 6546  df-fv 6552  df-fbas 21334  df-fil 23836

This theorem is referenced by:  isfil2  23846  filfi  23849  filinn0  23850  infil  23853  filconn  23873  filuni  23875  trfil2  23877  trfilss  23879  ufprim  23899  filufint  23910  rnelfmlem  23942  rnelfm  23943  fmfnfmlem2  23945  fmfnfmlem3  23946  fmfnfmlem4  23947  fmfnfm  23948  txflf  23996  fclsrest  24014  metust  24553  filnetlem3  36103