Theorem isfbas 23785

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 21318	. . . 4 ⊢ fBas = (𝑧 ∈ V ↦ {𝑤 ∈ 𝒫 𝒫 𝑧 ∣ (𝑤 ≠ ∅ ∧ ∅ ∉ 𝑤 ∧ ∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)})
2		neeq1 2995	. . . . . 6 ⊢ (𝑤 = 𝐹 → (𝑤 ≠ ∅ ↔ 𝐹 ≠ ∅))
3		neleq2 3044	. . . . . 6 ⊢ (𝑤 = 𝐹 → (∅ ∉ 𝑤 ↔ ∅ ∉ 𝐹))
4		ineq1 4167	. . . . . . . . 9 ⊢ (𝑤 = 𝐹 → (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) = (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)))
5	4	neeq1d 2992	. . . . . . . 8 ⊢ (𝑤 = 𝐹 → ((𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅ ↔ (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))
6	5	raleqbi1dv 3310	. . . . . . 7 ⊢ (𝑤 = 𝐹 → (∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅ ↔ ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))
7	6	raleqbi1dv 3310	. . . . . 6 ⊢ (𝑤 = 𝐹 → (∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑤 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅ ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))
8	2, 3, 7	3anbi123d 1439	. . . . 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 4570	. . . . 5 ⊢ (𝑧 = 𝐵 → 𝒫 𝑧 = 𝒫 𝐵)
11	10	pweqd 4573	. . . 4 ⊢ (𝑧 = 𝐵 → 𝒫 𝒫 𝑧 = 𝒫 𝒫 𝐵)
12		vpwex 5324	. . . . . 6 ⊢ 𝒫 𝑧 ∈ V
13	12	pwex 5327	. . . . 5 ⊢ 𝒫 𝒫 𝑧 ∈ V
14	13	a1i 11	. . . 4 ⊢ (𝑧 ∈ V → 𝒫 𝒫 𝑧 ∈ V)
15	1, 9, 11, 14	elmptrab 23783	. . 3 ⊢ (𝐹 ∈ (fBas‘𝐵) ↔ (𝐵 ∈ V ∧ 𝐹 ∈ 𝒫 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅)))
16		3anass 1095	. . 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 5325	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → 𝒫 𝐵 ∈ V)
19		elpw2g 5280	. . . . 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 632	. . 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 3463	. . . 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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ∉ wnel 3037  ∀wral 3052  Vcvv 3442   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  ‘cfv 6500  fBascfbas 21309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fv 6508  df-fbas 21318

This theorem is referenced by:  fbasne0  23786  0nelfb  23787  fbsspw  23788  isfbas2  23791  trfbas2  23799  fbasweak  23821  zfbas  23852  tsmsfbas  24084  ustfilxp  24169  minveclem3b  25396