Theorem isfbas 23653

Description: The predicate "𝐹 is a filter base." Note that some authors require filter bases to be closed under pairwise intersections, but that is not necessary under our definition. One advantage of this definition is that tails in a directed set form a filter base under our meaning. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)

Assertion

Ref	Expression
isfbas	⊢ (𝐵 ∈ 𝐴 → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))))

Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝐵,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem isfbas

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fbas 21230	. . . 4 ⊢ fBas = (𝑧 ∈ V ↦ {𝑤 ∈ 𝒫 𝒫 𝑧 ∣ (𝑤 ≠ ∅ ∧ ∅ ∉ 𝑤 ∧ ∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)})
2		neeq1 3002	. . . . . 6 ⊢ (𝑤 = 𝐹 → (𝑤 ≠ ∅ ↔ 𝐹 ≠ ∅))
3		neleq2 3052	. . . . . 6 ⊢ (𝑤 = 𝐹 → (∅ ∉ 𝑤 ↔ ∅ ∉ 𝐹))
4		ineq1 4205	. . . . . . . . 9 ⊢ (𝑤 = 𝐹 → (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) = (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)))
5	4	neeq1d 2999	. . . . . . . 8 ⊢ (𝑤 = 𝐹 → ((𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅ ↔ (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))
6	5	raleqbi1dv 3332	. . . . . . 7 ⊢ (𝑤 = 𝐹 → (∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅ ↔ ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))
7	6	raleqbi1dv 3332	. . . . . 6 ⊢ (𝑤 = 𝐹 → (∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅ ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))
8	2, 3, 7	3anbi123d 1435	. . . . 5 ⊢ (𝑤 = 𝐹 → ((𝑤 ≠ ∅ ∧ ∅ ∉ 𝑤 ∧ ∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅) ↔ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))
9	8	adantl 481	. . . 4 ⊢ ((𝑧 = 𝐵 ∧ 𝑤 = 𝐹) → ((𝑤 ≠ ∅ ∧ ∅ ∉ 𝑤 ∧ ∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅) ↔ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))
10		pweq 4616	. . . . 5 ⊢ (𝑧 = 𝐵 → 𝒫 𝑧 = 𝒫 𝐵)
11	10	pweqd 4619	. . . 4 ⊢ (𝑧 = 𝐵 → 𝒫 𝒫 𝑧 = 𝒫 𝒫 𝐵)
12		vpwex 5375	. . . . . 6 ⊢ 𝒫 𝑧 ∈ V
13	12	pwex 5378	. . . . 5 ⊢ 𝒫 𝒫 𝑧 ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝑧 ∈ V → 𝒫 𝒫 𝑧 ∈ V)
15	1, 9, 11, 14	elmptrab 23651	. . 3 ⊢ (𝐹 ∈ (fBas‘𝐵) ↔ (𝐵 ∈ V ∧ 𝐹 ∈ 𝒫 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))
16		3anass 1094	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝐹 ∈ 𝒫 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)) ↔ (𝐵 ∈ V ∧ (𝐹 ∈ 𝒫 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))))
17	15, 16	bitri 275	. 2 ⊢ (𝐹 ∈ (fBas‘𝐵) ↔ (𝐵 ∈ V ∧ (𝐹 ∈ 𝒫 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))))
18		pwexg 5376	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → 𝒫 𝐵 ∈ V)
19		elpw2g 5344	. . . . 5 ⊢ (𝒫 𝐵 ∈ V → (𝐹 ∈ 𝒫 𝒫 𝐵 ↔ 𝐹 ⊆ 𝒫 𝐵))
20	18, 19	syl 17	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐹 ∈ 𝒫 𝒫 𝐵 ↔ 𝐹 ⊆ 𝒫 𝐵))
21	20	anbi1d 629	. . 3 ⊢ (𝐵 ∈ 𝐴 → ((𝐹 ∈ 𝒫 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))))
22		elex 3492	. . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V)
23	22	biantrurd 532	. . 3 ⊢ (𝐵 ∈ 𝐴 → ((𝐹 ∈ 𝒫 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)) ↔ (𝐵 ∈ V ∧ (𝐹 ∈ 𝒫 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))))
24	21, 23	bitr3d 281	. 2 ⊢ (𝐵 ∈ 𝐴 → ((𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)) ↔ (𝐵 ∈ V ∧ (𝐹 ∈ 𝒫 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))))
25	17, 24	bitr4id 290	1 ⊢ (𝐵 ∈ 𝐴 → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2939   ∉ wnel 3045  ∀wral 3060  Vcvv 3473   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602  ‘cfv 6543  fBascfbas 21221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-fbas 21230

This theorem is referenced by:  fbasne0  23654  0nelfb  23655  fbsspw  23656  isfbas2  23659  trfbas2  23667  fbasweak  23689  zfbas  23720  tsmsfbas  23952  ustfilxp  24037  minveclem3b  25276