Theorem fsubbas 23832

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 23795	. . . . . 6 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → (fi‘𝐴) ≠ ∅)
2		fvprc 6832	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) = ∅)
3	2	necon1ai 2959	. . . . . 6 ⊢ ((fi‘𝐴) ≠ ∅ → 𝐴 ∈ V)
4	1, 3	syl 17	. . . . 5 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ∈ V)
5		ssfii 9332	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ⊆ (fi‘𝐴))
6	4, 5	syl 17	. . . 4 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ⊆ (fi‘𝐴))
7		fbsspw 23797	. . . 4 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋)
8	6, 7	sstrd 3932	. . 3 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ⊆ 𝒫 𝑋)
9		fieq0 9334	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅))
10	9	necon3bid 2976	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ≠ ∅ ↔ (fi‘𝐴) ≠ ∅))
11	10	biimpar 477	. . . 4 ⊢ ((𝐴 ∈ V ∧ (fi‘𝐴) ≠ ∅) → 𝐴 ≠ ∅)
12	4, 1, 11	syl2anc 585	. . 3 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ≠ ∅)
13		0nelfb 23796	. . 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 1196	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ⊆ 𝒫 𝑋)
16		fipwss 9342	. . . . 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 5320	. . . . . . . 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 5265	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ∈ V)
21		simpr2 1197	. . . . . 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 585	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → (fi‘𝐴) ≠ ∅)
24		simpr3 1198	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ¬ ∅ ∈ (fi‘𝐴))
25		df-nel 3037	. . . . . 6 ⊢ (∅ ∉ (fi‘𝐴) ↔ ¬ ∅ ∈ (fi‘𝐴))
26	24, 25	sylibr 234	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ∅ ∉ (fi‘𝐴))
27		fiin 9335	. . . . . . . 8 ⊢ ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥 ∩ 𝑦) ∈ (fi‘𝐴))
28		ssid 3944	. . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)
29		sseq1 3947	. . . . . . . . 9 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
30	29	rspcev 3564	. . . . . . . 8 ⊢ (((𝑥 ∩ 𝑦) ∈ (fi‘𝐴) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)) → ∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥 ∩ 𝑦))
31	27, 28, 30	sylancl 587	. . . . . . 7 ⊢ ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → ∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥 ∩ 𝑦))
32	31	rgen2 3177	. . . . . 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 23800	. . . . 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 714	. . 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 2114   ≠ wne 2932   ∉ wnel 3036  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  ‘cfv 6498  ficfi 9323  fBascfbas 21340

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 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-om 7818  df-1o 8405  df-2o 8406  df-en 8894  df-fin 8897  df-fi 9324  df-fbas 21349

This theorem is referenced by:  isufil2  23873  ufileu  23884  filufint  23885  fmfnfm  23923  hausflim  23946  flimclslem  23949  fclsfnflim  23992  flimfnfcls  23993  fclscmp  23995