Theorem fsubbas 23875

Description: A condition for a set to generate a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
fsubbas	⊢ (𝑋 ∈ 𝑉 → ((fi‘𝐴) ∈ (fBas‘𝑋) ↔ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))))

Proof of Theorem fsubbas

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fbasne0 23838	. . . . . 6 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → (fi‘𝐴) ≠ ∅)
2		fvprc 6898	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) = ∅)
3	2	necon1ai 2968	. . . . . 6 ⊢ ((fi‘𝐴) ≠ ∅ → 𝐴 ∈ V)
4	1, 3	syl 17	. . . . 5 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ∈ V)
5		ssfii 9459	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ⊆ (fi‘𝐴))
6	4, 5	syl 17	. . . 4 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ⊆ (fi‘𝐴))
7		fbsspw 23840	. . . 4 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋)
8	6, 7	sstrd 3994	. . 3 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ⊆ 𝒫 𝑋)
9		fieq0 9461	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅))
10	9	necon3bid 2985	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ≠ ∅ ↔ (fi‘𝐴) ≠ ∅))
11	10	biimpar 477	. . . 4 ⊢ ((𝐴 ∈ V ∧ (fi‘𝐴) ≠ ∅) → 𝐴 ≠ ∅)
12	4, 1, 11	syl2anc 584	. . 3 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ≠ ∅)
13		0nelfb 23839	. . 3 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐴))
14	8, 12, 13	3jca 1129	. 2 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴)))
15		simpr1 1195	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ⊆ 𝒫 𝑋)
16		fipwss 9469	. . . . 5 ⊢ (𝐴 ⊆ 𝒫 𝑋 → (fi‘𝐴) ⊆ 𝒫 𝑋)
17	15, 16	syl 17	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → (fi‘𝐴) ⊆ 𝒫 𝑋)
18		pwexg 5378	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
19	18	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝒫 𝑋 ∈ V)
20	19, 15	ssexd 5324	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ∈ V)
21		simpr2 1196	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ≠ ∅)
22	10	biimpa 476	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐴 ≠ ∅) → (fi‘𝐴) ≠ ∅)
23	20, 21, 22	syl2anc 584	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → (fi‘𝐴) ≠ ∅)
24		simpr3 1197	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ¬ ∅ ∈ (fi‘𝐴))
25		df-nel 3047	. . . . . 6 ⊢ (∅ ∉ (fi‘𝐴) ↔ ¬ ∅ ∈ (fi‘𝐴))
26	24, 25	sylibr 234	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ∅ ∉ (fi‘𝐴))
27		fiin 9462	. . . . . . . 8 ⊢ ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥 ∩ 𝑦) ∈ (fi‘𝐴))
28		ssid 4006	. . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)
29		sseq1 4009	. . . . . . . . 9 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
30	29	rspcev 3622	. . . . . . . 8 ⊢ (((𝑥 ∩ 𝑦) ∈ (fi‘𝐴) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)) → ∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥 ∩ 𝑦))
31	27, 28, 30	sylancl 586	. . . . . . 7 ⊢ ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → ∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥 ∩ 𝑦))
32	31	rgen2 3199	. . . . . 6 ⊢ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥 ∩ 𝑦)
33	32	a1i 11	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥 ∩ 𝑦))
34	23, 26, 33	3jca 1129	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ((fi‘𝐴) ≠ ∅ ∧ ∅ ∉ (fi‘𝐴) ∧ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥 ∩ 𝑦)))
35		isfbas2 23843	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → ((fi‘𝐴) ∈ (fBas‘𝑋) ↔ ((fi‘𝐴) ⊆ 𝒫 𝑋 ∧ ((fi‘𝐴) ≠ ∅ ∧ ∅ ∉ (fi‘𝐴) ∧ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥 ∩ 𝑦)))))
36	35	adantr 480	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ((fi‘𝐴) ∈ (fBas‘𝑋) ↔ ((fi‘𝐴) ⊆ 𝒫 𝑋 ∧ ((fi‘𝐴) ≠ ∅ ∧ ∅ ∉ (fi‘𝐴) ∧ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥 ∩ 𝑦)))))
37	17, 34, 36	mpbir2and 713	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → (fi‘𝐴) ∈ (fBas‘𝑋))
38	37	ex 412	. 2 ⊢ (𝑋 ∈ 𝑉 → ((𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴)) → (fi‘𝐴) ∈ (fBas‘𝑋)))
39	14, 38	impbid2 226	1 ⊢ (𝑋 ∈ 𝑉 → ((fi‘𝐴) ∈ (fBas‘𝑋) ↔ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108   ≠ wne 2940   ∉ wnel 3046  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  ‘cfv 6561  ficfi 9450  fBascfbas 21352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-2o 8507  df-en 8986  df-fin 8989  df-fi 9451  df-fbas 21361

This theorem is referenced by:  isufil2  23916  ufileu  23927  filufint  23928  fmfnfm  23966  hausflim  23989  flimclslem  23992  fclsfnflim  24035  flimfnfcls  24036  fclscmp  24038