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Theorem fsubbas 21872
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 21835 . . . . . 6 ((fi‘𝐴) ∈ (fBas‘𝑋) → (fi‘𝐴) ≠ ∅)
2 fvprc 6346 . . . . . . 7 𝐴 ∈ V → (fi‘𝐴) = ∅)
32necon1ai 2959 . . . . . 6 ((fi‘𝐴) ≠ ∅ → 𝐴 ∈ V)
41, 3syl 17 . . . . 5 ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ∈ V)
5 ssfii 8490 . . . . 5 (𝐴 ∈ V → 𝐴 ⊆ (fi‘𝐴))
64, 5syl 17 . . . 4 ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ⊆ (fi‘𝐴))
7 fbsspw 21837 . . . 4 ((fi‘𝐴) ∈ (fBas‘𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋)
86, 7sstrd 3754 . . 3 ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ⊆ 𝒫 𝑋)
9 fieq0 8492 . . . . . 6 (𝐴 ∈ V → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅))
109necon3bid 2976 . . . . 5 (𝐴 ∈ V → (𝐴 ≠ ∅ ↔ (fi‘𝐴) ≠ ∅))
1110biimpar 503 . . . 4 ((𝐴 ∈ V ∧ (fi‘𝐴) ≠ ∅) → 𝐴 ≠ ∅)
124, 1, 11syl2anc 696 . . 3 ((fi‘𝐴) ∈ (fBas‘𝑋) → 𝐴 ≠ ∅)
13 0nelfb 21836 . . 3 ((fi‘𝐴) ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐴))
148, 12, 133jca 1123 . 2 ((fi‘𝐴) ∈ (fBas‘𝑋) → (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴)))
15 simpr1 1234 . . . . 5 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ⊆ 𝒫 𝑋)
16 fipwss 8500 . . . . 5 (𝐴 ⊆ 𝒫 𝑋 → (fi‘𝐴) ⊆ 𝒫 𝑋)
1715, 16syl 17 . . . 4 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → (fi‘𝐴) ⊆ 𝒫 𝑋)
18 pwexg 4999 . . . . . . . 8 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
1918adantr 472 . . . . . . 7 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝒫 𝑋 ∈ V)
2019, 15ssexd 4957 . . . . . 6 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ∈ V)
21 simpr2 1236 . . . . . 6 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → 𝐴 ≠ ∅)
2210biimpa 502 . . . . . 6 ((𝐴 ∈ V ∧ 𝐴 ≠ ∅) → (fi‘𝐴) ≠ ∅)
2320, 21, 22syl2anc 696 . . . . 5 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → (fi‘𝐴) ≠ ∅)
24 simpr3 1238 . . . . . 6 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ¬ ∅ ∈ (fi‘𝐴))
25 df-nel 3036 . . . . . 6 (∅ ∉ (fi‘𝐴) ↔ ¬ ∅ ∈ (fi‘𝐴))
2624, 25sylibr 224 . . . . 5 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ∅ ∉ (fi‘𝐴))
27 fiin 8493 . . . . . . . 8 ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → (𝑥𝑦) ∈ (fi‘𝐴))
28 ssid 3765 . . . . . . . 8 (𝑥𝑦) ⊆ (𝑥𝑦)
29 sseq1 3767 . . . . . . . . 9 (𝑧 = (𝑥𝑦) → (𝑧 ⊆ (𝑥𝑦) ↔ (𝑥𝑦) ⊆ (𝑥𝑦)))
3029rspcev 3449 . . . . . . . 8 (((𝑥𝑦) ∈ (fi‘𝐴) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)) → ∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥𝑦))
3127, 28, 30sylancl 697 . . . . . . 7 ((𝑥 ∈ (fi‘𝐴) ∧ 𝑦 ∈ (fi‘𝐴)) → ∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥𝑦))
3231rgen2a 3115 . . . . . 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 1123 . . . 4 ((𝑋𝑉 ∧ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))) → ((fi‘𝐴) ≠ ∅ ∧ ∅ ∉ (fi‘𝐴) ∧ ∀𝑥 ∈ (fi‘𝐴)∀𝑦 ∈ (fi‘𝐴)∃𝑧 ∈ (fi‘𝐴)𝑧 ⊆ (𝑥𝑦)))
35 isfbas2 21840 . . . . 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 472 . . . 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 995 . . 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 1072  wcel 2139  wne 2932  wnel 3035  wral 3050  wrex 3051  Vcvv 3340  cin 3714  wss 3715  c0 4058  𝒫 cpw 4302  cfv 6049  ficfi 8481  fBascfbas 19936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-fin 8125  df-fi 8482  df-fbas 19945
This theorem is referenced by:  isufil2  21913  ufileu  21924  filufint  21925  fmfnfm  21963  hausflim  21986  flimclslem  21989  fclsfnflim  22032  flimfnfcls  22033  fclscmp  22035
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