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Theorem fsubbas 21718
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 21681 . . . . . 6 ((fi‘𝐴) ∈ (fBas‘𝑋) → (fi‘𝐴) ≠ ∅)
2 fvprc 6223 . . . . . . 7 𝐴 ∈ V → (fi‘𝐴) = ∅)
32necon1ai 2850 . . . . . 6 ((fi‘𝐴) ≠ ∅ → 𝐴 ∈ V)
41, 3syl 17 . . . . 5 ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ∈ V)
5 ssfii 8366 . . . . 5 (𝐴 ∈ V → 𝐴 ⊆ (fi‘𝐴))
64, 5syl 17 . . . 4 ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ⊆ (fi‘𝐴))
7 fbsspw 21683 . . . 4 ((fi‘𝐴) ∈ (fBas‘𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋)
86, 7sstrd 3646 . . 3 ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ⊆ 𝒫 𝑋)
9 fieq0 8368 . . . . . 6 (𝐴 ∈ V → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅))
109necon3bid 2867 . . . . 5 (𝐴 ∈ V → (𝐴 ≠ ∅ ↔ (fi‘𝐴) ≠ ∅))
1110biimpar 501 . . . 4 ((𝐴 ∈ V ∧ (fi‘𝐴) ≠ ∅) → 𝐴 ≠ ∅)
124, 1, 11syl2anc 694 . . 3 ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ≠ ∅)
13 0nelfb 21682 . . 3 ((fi‘𝐴) ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐴))
148, 12, 133jca 1261 . 2 ((fi‘𝐴) ∈ (fBas‘𝑋) → (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴)))
15 simpr1 1087 . . . . 5 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ⊆ 𝒫 𝑋)
16 fipwss 8376 . . . . 5 (𝐴 ⊆ 𝒫 𝑋 → (fi‘𝐴) ⊆ 𝒫 𝑋)
1715, 16syl 17 . . . 4 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → (fi‘𝐴) ⊆ 𝒫 𝑋)
18 pwexg 4880 . . . . . . . 8 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
1918adantr 480 . . . . . . 7 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝒫 𝑋 ∈ V)
2019, 15ssexd 4838 . . . . . 6 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ∈ V)
21 simpr2 1088 . . . . . 6 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ≠ ∅)
2210biimpa 500 . . . . . 6 ((𝐴 ∈ V ∧ 𝐴 ≠ ∅) → (fi‘𝐴) ≠ ∅)
2320, 21, 22syl2anc 694 . . . . 5 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → (fi‘𝐴) ≠ ∅)
24 simpr3 1089 . . . . . 6 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ¬ ∅ ∈ (fi‘𝐴))
25 df-nel 2927 . . . . . 6 (∅ ∉ (fi‘𝐴) ↔ ¬ ∅ ∈ (fi‘𝐴))
2624, 25sylibr 224 . . . . 5 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ∅ ∉ (fi‘𝐴))
27 fiin 8369 . . . . . . . 8 ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥𝑦) ∈ (fi‘𝐴))
28 ssid 3657 . . . . . . . 8 (𝑥𝑦) ⊆ (𝑥𝑦)
29 sseq1 3659 . . . . . . . . 9 (𝑧 = (𝑥𝑦) → (𝑧 ⊆ (𝑥𝑦) ↔ (𝑥𝑦) ⊆ (𝑥𝑦)))
3029rspcev 3340 . . . . . . . 8 (((𝑥𝑦) ∈ (fi‘𝐴) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)) → ∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥𝑦))
3127, 28, 30sylancl 695 . . . . . . 7 ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → ∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥𝑦))
3231rgen2a 3006 . . . . . 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 1261 . . . 4 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ((fi‘𝐴) ≠ ∅ ∧ ∅ ∉ (fi‘𝐴) ∧ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥𝑦)))
35 isfbas2 21686 . . . . 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 977 . . 3 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → (fi‘𝐴) ∈ (fBas‘𝑋))
3837ex 449 . 2 (𝑋𝑉 → ((𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴)) → (fi‘𝐴) ∈ (fBas‘𝑋)))
3914, 38impbid2 216 1 (𝑋𝑉 → ((fi‘𝐴) ∈ (fBas‘𝑋) ↔ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054  wcel 2030  wne 2823  wnel 2926  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  cfv 5926  ficfi 8357  fBascfbas 19782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-fin 8001  df-fi 8358  df-fbas 19791
This theorem is referenced by:  isufil2  21759  ufileu  21770  filufint  21771  fmfnfm  21809  hausflim  21832  flimclslem  21835  fclsfnflim  21878  flimfnfcls  21879  fclscmp  21881
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