Theorem fsubbas 23780

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 23743	. . . . . 6 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → (fi‘𝐴) ≠ ∅)
2		fvprc 6814	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) = ∅)
3	2	necon1ai 2955	. . . . . 6 ⊢ ((fi‘𝐴) ≠ ∅ → 𝐴 ∈ V)
4	1, 3	syl 17	. . . . 5 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ∈ V)
5		ssfii 9303	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ⊆ (fi‘𝐴))
6	4, 5	syl 17	. . . 4 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ⊆ (fi‘𝐴))
7		fbsspw 23745	. . . 4 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋)
8	6, 7	sstrd 3945	. . 3 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ⊆ 𝒫 𝑋)
9		fieq0 9305	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅))
10	9	necon3bid 2972	. . . . 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 23744	. . 3 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐴))
14	8, 12, 13	3jca 1128	. 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 9313	. . . . 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 5316	. . . . . . . 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 5262	. . . . . 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 3033	. . . . . 6 ⊢ (∅ ∉ (fi‘𝐴) ↔ ¬ ∅ ∈ (fi‘𝐴))
26	24, 25	sylibr 234	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ∅ ∉ (fi‘𝐴))
27		fiin 9306	. . . . . . . 8 ⊢ ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥 ∩ 𝑦) ∈ (fi‘𝐴))
28		ssid 3957	. . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)
29		sseq1 3960	. . . . . . . . 9 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
30	29	rspcev 3577	. . . . . . . 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 3172	. . . . . 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 1128	. . . 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 23748	. . . . 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 1086   ∈ wcel 2111   ≠ wne 2928   ∉ wnel 3032  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  𝒫 cpw 4550  ‘cfv 6481  ficfi 9294  fBascfbas 21277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-om 7797  df-1o 8385  df-2o 8386  df-en 8870  df-fin 8873  df-fi 9295  df-fbas 21286

This theorem is referenced by:  isufil2  23821  ufileu  23832  filufint  23833  fmfnfm  23871  hausflim  23894  flimclslem  23897  fclsfnflim  23940  flimfnfcls  23941  fclscmp  23943