Theorem isfbas 24128

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 21655	. . . 4 ⊢ fBas = (𝑧 ∈ V ↦ {𝑤 ∈ 𝒫 𝒫 𝑧 ∣ (𝑤 ≠ ∅ ∧ ∅ ∉ 𝑤 ∧ ∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)})
2		neeq1 3018	. . . . . 6 ⊢ (𝑤 = 𝐹 → (𝑤 ≠ ∅ ↔ 𝐹 ≠ ∅))
3		neleq2 3069	. . . . . 6 ⊢ (𝑤 = 𝐹 → (∅ ∉ 𝑤 ↔ ∅ ∉ 𝐹))
4		ineq1 4159	. . . . . . . . 9 ⊢ (𝑤 = 𝐹 → (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) = (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)))
5	4	neeq1d 3015	. . . . . . . 8 ⊢ (𝑤 = 𝐹 → ((𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅ ↔ (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))
6	5	raleqbi1dv 3330	. . . . . . 7 ⊢ (𝑤 = 𝐹 → (∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅ ↔ ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))
7	6	raleqbi1dv 3330	. . . . . 6 ⊢ (𝑤 = 𝐹 → (∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅ ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))
8	2, 3, 7	3anbi123d 1464	. . . . 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 487	. . . 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 4571	. . . . 5 ⊢ (𝑧 = 𝐵 → 𝒫 𝑧 = 𝒫 𝐵)
11	10	pweqd 4574	. . . 4 ⊢ (𝑧 = 𝐵 → 𝒫 𝒫 𝑧 = 𝒫 𝒫 𝐵)
12		vpwex 5339	. . . . . 6 ⊢ 𝒫 𝑧 ∈ V
13	12	pwex 5342	. . . . 5 ⊢ 𝒫 𝒫 𝑧 ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝑧 ∈ V → 𝒫 𝒫 𝑧 ∈ V)
15	1, 9, 11, 14	elmptrab 24126	. . 3 ⊢ (𝐹 ∈ (fBas‘𝐵) ↔ (𝐵 ∈ V ∧ 𝐹 ∈ 𝒫 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))
16		3anass 1111	. . 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 278	. 2 ⊢ (𝐹 ∈ (fBas‘𝐵) ↔ (𝐵 ∈ V ∧ (𝐹 ∈ 𝒫 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))))
18		pwexg 5340	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → 𝒫 𝐵 ∈ V)
19		elpw2g 5295	. . . . 5 ⊢ (𝒫 𝐵 ∈ V → (𝐹 ∈ 𝒫 𝒫 𝐵 ↔ 𝐹 ⊆ 𝒫 𝐵))
20	18, 19	syl 18	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐹 ∈ 𝒫 𝒫 𝐵 ↔ 𝐹 ⊆ 𝒫 𝐵))
21	20	anbi1d 643	. . 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 3472	. . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V)
23	22	biantrurd 542	. . 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 284	. 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 293	1 ⊢ (𝐵 ∈ 𝐴 → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956   ∉ wnel 3062  ∀wral 3077  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  ‘cfv 6531  fBascfbas 21646

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fv 6539  df-fbas 21655

This theorem is used by:  fbasne0  24129  0nelfb  24130  fbsspw  24131  isfbas2  24134  trfbas2  24142  fbasweak  24164  zfbas  24195  tsmsfbas  24427  ustfilxp  24512  minveclem3b  25729