Theorem filin 22465

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 22459	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2		fbasssin 22447	. . 3 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵))
3	1, 2	syl3an1 1159	. 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵))
4		inss1 4208	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
5		filelss 22463	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋)
6	4, 5	sstrid 3981	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐴 ∩ 𝐵) ⊆ 𝑋)
7		filss 22464	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ (𝐴 ∩ 𝐵) ⊆ 𝑋 ∧ 𝑥 ⊆ (𝐴 ∩ 𝐵))) → (𝐴 ∩ 𝐵) ∈ 𝐹)
8	7	3exp2 1350	. . . . . . 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 409	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∩ 𝐵) ⊆ 𝑋) → (𝑥 ∈ 𝐹 → (𝑥 ⊆ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐹)))
11	10	rexlimdv 3286	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∩ 𝐵) ⊆ 𝑋) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐹))
12	6, 11	syldan 593	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐹))
13	12	3adant3 1128	. 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 398   ∧ w3a 1083   ∈ wcel 2113  ∃wrex 3142   ∩ cin 3938   ⊆ wss 3939  ‘cfv 6358  fBascfbas 20536  Filcfil 22456

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fv 6366  df-fbas 20545  df-fil 22457

This theorem is referenced by:  isfil2  22467  filfi  22470  filinn0  22471  infil  22474  filconn  22494  filuni  22496  trfil2  22498  trfilss  22500  ufprim  22520  filufint  22531  rnelfmlem  22563  rnelfm  22564  fmfnfmlem2  22566  fmfnfmlem3  22567  fmfnfmlem4  22568  fmfnfm  22569  txflf  22617  fclsrest  22635  metust  23171  filnetlem3  33732