Theorem isfbas 22041

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		pwexg 5090	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → 𝒫 𝐵 ∈ V)
2		elpw2g 5061	. . . . 5 ⊢ (𝒫 𝐵 ∈ V → (𝐹 ∈ 𝒫 𝒫 𝐵 ↔ 𝐹 ⊆ 𝒫 𝐵))
3	1, 2	syl 17	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐹 ∈ 𝒫 𝒫 𝐵 ↔ 𝐹 ⊆ 𝒫 𝐵))
4	3	anbi1d 623	. . 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)) ≠ ∅))))
5		elex 3413	. . . 4 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ∈ V)
6	5	biantrurd 528	. . 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)) ≠ ∅)))))
7	4, 6	bitr3d 273	. 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)) ≠ ∅)))))
8		df-fbas 20139	. . . 4 ⊢ fBas = (𝑧 ∈ V ↦ {𝑤 ∈ 𝒫 𝒫 𝑧 ∣ (𝑤 ≠ ∅ ∧ ∅ ∉ 𝑤 ∧ ∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)})
9		neeq1 3030	. . . . . 6 ⊢ (𝑤 = 𝐹 → (𝑤 ≠ ∅ ↔ 𝐹 ≠ ∅))
10		neleq2 3080	. . . . . 6 ⊢ (𝑤 = 𝐹 → (∅ ∉ 𝑤 ↔ ∅ ∉ 𝐹))
11		ineq1 4029	. . . . . . . . 9 ⊢ (𝑤 = 𝐹 → (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) = (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)))
12	11	neeq1d 3027	. . . . . . . 8 ⊢ (𝑤 = 𝐹 → ((𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅ ↔ (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))
13	12	raleqbi1dv 3327	. . . . . . 7 ⊢ (𝑤 = 𝐹 → (∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅ ↔ ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))
14	13	raleqbi1dv 3327	. . . . . 6 ⊢ (𝑤 = 𝐹 → (∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅ ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))
15	9, 10, 14	3anbi123d 1509	. . . . 5 ⊢ (𝑤 = 𝐹 → ((𝑤 ≠ ∅ ∧ ∅ ∉ 𝑤 ∧ ∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅) ↔ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))
16	15	adantl 475	. . . 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)) ≠ ∅)))
17		pweq 4381	. . . . 5 ⊢ (𝑧 = 𝐵 → 𝒫 𝑧 = 𝒫 𝐵)
18	17	pweqd 4383	. . . 4 ⊢ (𝑧 = 𝐵 → 𝒫 𝒫 𝑧 = 𝒫 𝒫 𝐵)
19		vpwex 5089	. . . . . 6 ⊢ 𝒫 𝑧 ∈ V
20	19	pwex 5092	. . . . 5 ⊢ 𝒫 𝒫 𝑧 ∈ V
21	20	a1i 11	. . . 4 ⊢ (𝑧 ∈ V → 𝒫 𝒫 𝑧 ∈ V)
22	8, 16, 18, 21	elmptrab 22039	. . 3 ⊢ (𝐹 ∈ (fBas‘𝐵) ↔ (𝐵 ∈ V ∧ 𝐹 ∈ 𝒫 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))
23		3anass 1079	. . 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)) ≠ ∅))))
24	22, 23	bitri 267	. 2 ⊢ (𝐹 ∈ (fBas‘𝐵) ↔ (𝐵 ∈ V ∧ (𝐹 ∈ 𝒫 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))))
25	7, 24	syl6rbbr 282	1 ⊢ (𝐵 ∈ 𝐴 → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106   ≠ wne 2968   ∉ wnel 3074  ∀wral 3089  Vcvv 3397   ∩ cin 3790   ⊆ wss 3791  ∅c0 4140  𝒫 cpw 4378  ‘cfv 6135  fBascfbas 20130

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 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138

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 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fv 6143  df-fbas 20139

This theorem is referenced by:  fbasne0  22042  0nelfb  22043  fbsspw  22044  isfbas2  22047  trfbas2  22055  fbasweak  22077  zfbas  22108  tsmsfbas  22339  ustfilxp  22424  minveclem3b  23634