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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 (𝑋𝑉 → ((fi‘𝐴) ∈ (fBas‘𝑋) ↔ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))))

Proof of Theorem fsubbas
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fbasne0 23838 . . . . . 6 ((fi‘𝐴) ∈ (fBas‘𝑋) → (fi‘𝐴) ≠ ∅)
2 fvprc 6898 . . . . . . 7 𝐴 ∈ V → (fi‘𝐴) = ∅)
32necon1ai 2968 . . . . . 6 ((fi‘𝐴) ≠ ∅ → 𝐴 ∈ V)
41, 3syl 17 . . . . 5 ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ∈ V)
5 ssfii 9459 . . . . 5 (𝐴 ∈ V → 𝐴 ⊆ (fi‘𝐴))
64, 5syl 17 . . . 4 ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ⊆ (fi‘𝐴))
7 fbsspw 23840 . . . 4 ((fi‘𝐴) ∈ (fBas‘𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋)
86, 7sstrd 3994 . . 3 ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ⊆ 𝒫 𝑋)
9 fieq0 9461 . . . . . 6 (𝐴 ∈ V → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅))
109necon3bid 2985 . . . . 5 (𝐴 ∈ V → (𝐴 ≠ ∅ ↔ (fi‘𝐴) ≠ ∅))
1110biimpar 477 . . . 4 ((𝐴 ∈ V ∧ (fi‘𝐴) ≠ ∅) → 𝐴 ≠ ∅)
124, 1, 11syl2anc 584 . . 3 ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ≠ ∅)
13 0nelfb 23839 . . 3 ((fi‘𝐴) ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐴))
148, 12, 133jca 1129 . 2 ((fi‘𝐴) ∈ (fBas‘𝑋) → (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴)))
15 simpr1 1195 . . . . 5 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ⊆ 𝒫 𝑋)
16 fipwss 9469 . . . . 5 (𝐴 ⊆ 𝒫 𝑋 → (fi‘𝐴) ⊆ 𝒫 𝑋)
1715, 16syl 17 . . . 4 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → (fi‘𝐴) ⊆ 𝒫 𝑋)
18 pwexg 5378 . . . . . . . 8 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
1918adantr 480 . . . . . . 7 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝒫 𝑋 ∈ V)
2019, 15ssexd 5324 . . . . . 6 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ∈ V)
21 simpr2 1196 . . . . . 6 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ≠ ∅)
2210biimpa 476 . . . . . 6 ((𝐴 ∈ V ∧ 𝐴 ≠ ∅) → (fi‘𝐴) ≠ ∅)
2320, 21, 22syl2anc 584 . . . . 5 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → (fi‘𝐴) ≠ ∅)
24 simpr3 1197 . . . . . 6 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ¬ ∅ ∈ (fi‘𝐴))
25 df-nel 3047 . . . . . 6 (∅ ∉ (fi‘𝐴) ↔ ¬ ∅ ∈ (fi‘𝐴))
2624, 25sylibr 234 . . . . 5 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ∅ ∉ (fi‘𝐴))
27 fiin 9462 . . . . . . . 8 ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥𝑦) ∈ (fi‘𝐴))
28 ssid 4006 . . . . . . . 8 (𝑥𝑦) ⊆ (𝑥𝑦)
29 sseq1 4009 . . . . . . . . 9 (𝑧 = (𝑥𝑦) → (𝑧 ⊆ (𝑥𝑦) ↔ (𝑥𝑦) ⊆ (𝑥𝑦)))
3029rspcev 3622 . . . . . . . 8 (((𝑥𝑦) ∈ (fi‘𝐴) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)) → ∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥𝑦))
3127, 28, 30sylancl 586 . . . . . . 7 ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → ∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥𝑦))
3231rgen2 3199 . . . . . 6 𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥𝑦)
3332a1i 11 . . . . 5 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥𝑦))
3423, 26, 333jca 1129 . . . 4 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ((fi‘𝐴) ≠ ∅ ∧ ∅ ∉ (fi‘𝐴) ∧ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥𝑦)))
35 isfbas2 23843 . . . . 5 (𝑋𝑉 → ((fi‘𝐴) ∈ (fBas‘𝑋) ↔ ((fi‘𝐴) ⊆ 𝒫 𝑋 ∧ ((fi‘𝐴) ≠ ∅ ∧ ∅ ∉ (fi‘𝐴) ∧ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥𝑦)))))
3635adantr 480 . . . 4 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ((fi‘𝐴) ∈ (fBas‘𝑋) ↔ ((fi‘𝐴) ⊆ 𝒫 𝑋 ∧ ((fi‘𝐴) ≠ ∅ ∧ ∅ ∉ (fi‘𝐴) ∧ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥𝑦)))))
3717, 34, 36mpbir2and 713 . . 3 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → (fi‘𝐴) ∈ (fBas‘𝑋))
3837ex 412 . 2 (𝑋𝑉 → ((𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴)) → (fi‘𝐴) ∈ (fBas‘𝑋)))
3914, 38impbid2 226 1 (𝑋𝑉 → ((fi‘𝐴) ∈ (fBas‘𝑋) ↔ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))))
Colors of variables: wff setvar class
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
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