MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsubbas Structured version   Visualization version   GIF version

Theorem fsubbas 21576
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 21539 . . . . . 6 ((fi‘𝐴) ∈ (fBas‘𝑋) → (fi‘𝐴) ≠ ∅)
2 fvprc 6144 . . . . . . 7 𝐴 ∈ V → (fi‘𝐴) = ∅)
32necon1ai 2823 . . . . . 6 ((fi‘𝐴) ≠ ∅ → 𝐴 ∈ V)
41, 3syl 17 . . . . 5 ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ∈ V)
5 ssfii 8270 . . . . 5 (𝐴 ∈ V → 𝐴 ⊆ (fi‘𝐴))
64, 5syl 17 . . . 4 ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ⊆ (fi‘𝐴))
7 fbsspw 21541 . . . 4 ((fi‘𝐴) ∈ (fBas‘𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋)
86, 7sstrd 3598 . . 3 ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ⊆ 𝒫 𝑋)
9 fieq0 8272 . . . . . 6 (𝐴 ∈ V → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅))
109necon3bid 2840 . . . . 5 (𝐴 ∈ V → (𝐴 ≠ ∅ ↔ (fi‘𝐴) ≠ ∅))
1110biimpar 502 . . . 4 ((𝐴 ∈ V ∧ (fi‘𝐴) ≠ ∅) → 𝐴 ≠ ∅)
124, 1, 11syl2anc 692 . . 3 ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ≠ ∅)
13 0nelfb 21540 . . 3 ((fi‘𝐴) ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐴))
148, 12, 133jca 1240 . 2 ((fi‘𝐴) ∈ (fBas‘𝑋) → (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴)))
15 simpr1 1065 . . . . 5 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ⊆ 𝒫 𝑋)
16 fipwss 8280 . . . . 5 (𝐴 ⊆ 𝒫 𝑋 → (fi‘𝐴) ⊆ 𝒫 𝑋)
1715, 16syl 17 . . . 4 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → (fi‘𝐴) ⊆ 𝒫 𝑋)
18 pwexg 4815 . . . . . . . 8 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
1918adantr 481 . . . . . . 7 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝒫 𝑋 ∈ V)
2019, 15ssexd 4770 . . . . . 6 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ∈ V)
21 simpr2 1066 . . . . . 6 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ≠ ∅)
2210biimpa 501 . . . . . 6 ((𝐴 ∈ V ∧ 𝐴 ≠ ∅) → (fi‘𝐴) ≠ ∅)
2320, 21, 22syl2anc 692 . . . . 5 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → (fi‘𝐴) ≠ ∅)
24 simpr3 1067 . . . . . 6 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ¬ ∅ ∈ (fi‘𝐴))
25 df-nel 2900 . . . . . 6 (∅ ∉ (fi‘𝐴) ↔ ¬ ∅ ∈ (fi‘𝐴))
2624, 25sylibr 224 . . . . 5 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ∅ ∉ (fi‘𝐴))
27 fiin 8273 . . . . . . . 8 ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥𝑦) ∈ (fi‘𝐴))
28 ssid 3608 . . . . . . . 8 (𝑥𝑦) ⊆ (𝑥𝑦)
29 sseq1 3610 . . . . . . . . 9 (𝑧 = (𝑥𝑦) → (𝑧 ⊆ (𝑥𝑦) ↔ (𝑥𝑦) ⊆ (𝑥𝑦)))
3029rspcev 3300 . . . . . . . 8 (((𝑥𝑦) ∈ (fi‘𝐴) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)) → ∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥𝑦))
3127, 28, 30sylancl 693 . . . . . . 7 ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → ∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥𝑦))
3231rgen2a 2976 . . . . . 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 1240 . . . 4 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ((fi‘𝐴) ≠ ∅ ∧ ∅ ∉ (fi‘𝐴) ∧ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥𝑦)))
35 isfbas2 21544 . . . . 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 481 . . . 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 956 . . 3 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → (fi‘𝐴) ∈ (fBas‘𝑋))
3837ex 450 . 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 384  w3a 1036  wcel 1992  wne 2796  wnel 2899  wral 2912  wrex 2913  Vcvv 3191  cin 3559  wss 3560  c0 3896  𝒫 cpw 4135  cfv 5850  ficfi 8261  fBascfbas 19648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-fin 7904  df-fi 8262  df-fbas 19657
This theorem is referenced by:  isufil2  21617  ufileu  21628  filufint  21629  fmfnfm  21667  hausflim  21690  flimclslem  21693  fclsfnflim  21736  flimfnfcls  21737  fclscmp  21739
  Copyright terms: Public domain W3C validator