Theorem fsubbas 23016

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 22979	. . . . . 6 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → (fi‘𝐴) ≠ ∅)
2		fvprc 6763	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (fi‘𝐴) = ∅)
3	2	necon1ai 2973	. . . . . 6 ⊢ ((fi‘𝐴) ≠ ∅ → 𝐴 ∈ V)
4	1, 3	syl 17	. . . . 5 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ∈ V)
5		ssfii 9156	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ⊆ (fi‘𝐴))
6	4, 5	syl 17	. . . 4 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ⊆ (fi‘𝐴))
7		fbsspw 22981	. . . 4 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋)
8	6, 7	sstrd 3936	. . 3 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ⊆ 𝒫 𝑋)
9		fieq0 9158	. . . . . 6 ⊢ (𝐴 ∈ V → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅))
10	9	necon3bid 2990	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ≠ ∅ ↔ (fi‘𝐴) ≠ ∅))
11	10	biimpar 478	. . . 4 ⊢ ((𝐴 ∈ V ∧ (fi‘𝐴) ≠ ∅) → 𝐴 ≠ ∅)
12	4, 1, 11	syl2anc 584	. . 3 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ≠ ∅)
13		0nelfb 22980	. . 3 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐴))
14	8, 12, 13	3jca 1127	. 2 ⊢ ((fi‘𝐴) ∈ (fBas‘𝑋) → (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴)))
15		simpr1 1193	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ⊆ 𝒫 𝑋)
16		fipwss 9166	. . . . 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 5305	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
19	18	adantr 481	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝒫 𝑋 ∈ V)
20	19, 15	ssexd 5252	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ∈ V)
21		simpr2 1194	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ≠ ∅)
22	10	biimpa 477	. . . . . 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 1195	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ¬ ∅ ∈ (fi‘𝐴))
25		df-nel 3052	. . . . . 6 ⊢ (∅ ∉ (fi‘𝐴) ↔ ¬ ∅ ∈ (fi‘𝐴))
26	24, 25	sylibr 233	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ∅ ∉ (fi‘𝐴))
27		fiin 9159	. . . . . . . 8 ⊢ ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥 ∩ 𝑦) ∈ (fi‘𝐴))
28		ssid 3948	. . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)
29		sseq1 3951	. . . . . . . . 9 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
30	29	rspcev 3561	. . . . . . . 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 3129	. . . . . 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 1127	. . . 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 22984	. . . . 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 481	. . . 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 710	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → (fi‘𝐴) ∈ (fBas‘𝑋))
38	37	ex 413	. 2 ⊢ (𝑋 ∈ 𝑉 → ((𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴)) → (fi‘𝐴) ∈ (fBas‘𝑋)))
39	14, 38	impbid2 225	1 ⊢ (𝑋 ∈ 𝑉 → ((fi‘𝐴) ∈ (fBas‘𝑋) ↔ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   ∈ wcel 2110   ≠ wne 2945   ∉ wnel 3051  ∀wral 3066  ∃wrex 3067  Vcvv 3431   ∩ cin 3891   ⊆ wss 3892  ∅c0 4262  𝒫 cpw 4539  ‘cfv 6432  ficfi 9147  fBascfbas 20583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-om 7707  df-1o 8288  df-er 8481  df-en 8717  df-fin 8720  df-fi 9148  df-fbas 20592

This theorem is referenced by:  isufil2  23057  ufileu  23068  filufint  23069  fmfnfm  23107  hausflim  23130  flimclslem  23133  fclsfnflim  23176  flimfnfcls  23177  fclscmp  23179