Theorem filin 23883

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 23877	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2		fbasssin 23865	. . 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 4258	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
5		filelss 23881	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋)
6	4, 5	sstrid 4020	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → (𝐴 ∩ 𝐵) ⊆ 𝑋)
7		filss 23882	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ (𝐴 ∩ 𝐵) ⊆ 𝑋 ∧ 𝑥 ⊆ (𝐴 ∩ 𝐵))) → (𝐴 ∩ 𝐵) ∈ 𝐹)
8	7	3exp2 1354	. . . . . . 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 3159	. . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∩ 𝐵) ⊆ 𝑋) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵) → (𝐴 ∩ 𝐵) ∈ 𝐹))
12	6, 11	syldan 590	. . 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 1087   ∈ wcel 2108  ∃wrex 3076   ∩ cin 3975   ⊆ wss 3976  ‘cfv 6573  fBascfbas 21375  Filcfil 23874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-fbas 21384  df-fil 23875

This theorem is referenced by:  isfil2  23885  filfi  23888  filinn0  23889  infil  23892  filconn  23912  filuni  23914  trfil2  23916  trfilss  23918  ufprim  23938  filufint  23949  rnelfmlem  23981  rnelfm  23982  fmfnfmlem2  23984  fmfnfmlem3  23985  fmfnfmlem4  23986  fmfnfm  23987  txflf  24035  fclsrest  24053  metust  24592  filnetlem3  36346