Theorem isfbas 24023

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 21556	. . . 4 ⊢ fBas = (𝑧 ∈ V ↦ {𝑤 ∈ 𝒫 𝒫 𝑧 ∣ (𝑤 ≠ ∅ ∧ ∅ ∉ 𝑤 ∧ ∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)})
2		neeq1 3023	. . . . . 6 ⊢ (𝑤 = 𝐹 → (𝑤 ≠ ∅ ↔ 𝐹 ≠ ∅))
3		neleq2 3074	. . . . . 6 ⊢ (𝑤 = 𝐹 → (∅ ∉ 𝑤 ↔ ∅ ∉ 𝐹))
4		ineq1 4169	. . . . . . . . 9 ⊢ (𝑤 = 𝐹 → (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) = (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)))
5	4	neeq1d 3020	. . . . . . . 8 ⊢ (𝑤 = 𝐹 → ((𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅ ↔ (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))
6	5	raleqbi1dv 3336	. . . . . . 7 ⊢ (𝑤 = 𝐹 → (∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅ ↔ ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))
7	6	raleqbi1dv 3336	. . . . . 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 4581	. . . . 5 ⊢ (𝑧 = 𝐵 → 𝒫 𝑧 = 𝒫 𝐵)
11	10	pweqd 4584	. . . 4 ⊢ (𝑧 = 𝐵 → 𝒫 𝒫 𝑧 = 𝒫 𝒫 𝐵)
12		vpwex 5353	. . . . . 6 ⊢ 𝒫 𝑧 ∈ V
13	12	pwex 5356	. . . . 5 ⊢ 𝒫 𝒫 𝑧 ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝑧 ∈ V → 𝒫 𝒫 𝑧 ∈ V)
15	1, 9, 11, 14	elmptrab 24021	. . 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 5354	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → 𝒫 𝐵 ∈ V)
19		elpw2g 5309	. . . . 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 3479	. . . 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 2146   ≠ wne 2961   ∉ wnel 3067  ∀wral 3082  Vcvv 3458   ∩ cin 3907   ⊆ wss 3908  ∅c0 4289  𝒫 cpw 4567  ‘cfv 6543  fBascfbas 21547

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fv 6551  df-fbas 21556

This theorem is used by:  fbasne0  24024  0nelfb  24025  fbsspw  24026  isfbas2  24029  trfbas2  24037  fbasweak  24059  zfbas  24090  tsmsfbas  24322  ustfilxp  24407  minveclem3b  25624