Theorem filin 22077

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 22071	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2		fbasssin 22059	. . 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 4053	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
5		filelss 22075	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋)
6	4, 5	syl5ss 3832	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐴 ∩ 𝐵) ⊆ 𝑋)
7		filss 22076	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ (𝐴 ∩ 𝐵) ⊆ 𝑋 ∧ 𝑥 ⊆ (𝐴 ∩ 𝐵))) → (𝐴 ∩ 𝐵) ∈ 𝐹)
8	7	3exp2 1416	. . . . . . 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 397	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∩ 𝐵) ⊆ 𝑋) → (𝑥 ∈ 𝐹 → (𝑥 ⊆ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐹)))
11	10	rexlimdv 3212	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∩ 𝐵) ⊆ 𝑋) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐹))
12	6, 11	syldan 585	. . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐹))
13	12	3adant3 1123	. 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 386   ∧ w3a 1071   ∈ wcel 2107  ∃wrex 3091   ∩ cin 3791   ⊆ wss 3792  ‘cfv 6137  fBascfbas 20141  Filcfil 22068

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

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  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-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  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-iota 6101  df-fun 6139  df-fv 6145  df-fbas 20150  df-fil 22069

This theorem is referenced by:  isfil2  22079  filfi  22082  filinn0  22083  infil  22086  filconn  22106  filuni  22108  trfil2  22110  trfilss  22112  ufprim  22132  filufint  22143  rnelfmlem  22175  rnelfm  22176  fmfnfmlem2  22178  fmfnfmlem3  22179  fmfnfmlem4  22180  fmfnfm  22181  txflf  22229  fclsrest  22247  metust  22782  filnetlem3  32971