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Theorem eltg3 22112
Description: Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
eltg3 (𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥𝐵𝐴 = 𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑉

Proof of Theorem eltg3
StepHypRef Expression
1 elfvdm 6806 . . . 4 (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen)
2 inex1g 5243 . . . 4 (𝐵 ∈ dom topGen → (𝐵 ∩ 𝒫 𝐴) ∈ V)
31, 2syl 17 . . 3 (𝐴 ∈ (topGen‘𝐵) → (𝐵 ∩ 𝒫 𝐴) ∈ V)
4 eltg4i 22110 . . 3 (𝐴 ∈ (topGen‘𝐵) → 𝐴 = (𝐵 ∩ 𝒫 𝐴))
5 inss1 4162 . . . . . 6 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐵
6 sseq1 3946 . . . . . 6 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → (𝑥𝐵 ↔ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐵))
75, 6mpbiri 257 . . . . 5 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → 𝑥𝐵)
87biantrurd 533 . . . 4 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → (𝐴 = 𝑥 ↔ (𝑥𝐵𝐴 = 𝑥)))
9 unieq 4850 . . . . 5 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → 𝑥 = (𝐵 ∩ 𝒫 𝐴))
109eqeq2d 2749 . . . 4 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → (𝐴 = 𝑥𝐴 = (𝐵 ∩ 𝒫 𝐴)))
118, 10bitr3d 280 . . 3 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → ((𝑥𝐵𝐴 = 𝑥) ↔ 𝐴 = (𝐵 ∩ 𝒫 𝐴)))
123, 4, 11spcedv 3537 . 2 (𝐴 ∈ (topGen‘𝐵) → ∃𝑥(𝑥𝐵𝐴 = 𝑥))
13 eltg3i 22111 . . . . 5 ((𝐵𝑉𝑥𝐵) → 𝑥 ∈ (topGen‘𝐵))
14 eleq1 2826 . . . . 5 (𝐴 = 𝑥 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝑥 ∈ (topGen‘𝐵)))
1513, 14syl5ibrcom 246 . . . 4 ((𝐵𝑉𝑥𝐵) → (𝐴 = 𝑥𝐴 ∈ (topGen‘𝐵)))
1615expimpd 454 . . 3 (𝐵𝑉 → ((𝑥𝐵𝐴 = 𝑥) → 𝐴 ∈ (topGen‘𝐵)))
1716exlimdv 1936 . 2 (𝐵𝑉 → (∃𝑥(𝑥𝐵𝐴 = 𝑥) → 𝐴 ∈ (topGen‘𝐵)))
1812, 17impbid2 225 1 (𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥𝐵𝐴 = 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wex 1782  wcel 2106  Vcvv 3432  cin 3886  wss 3887  𝒫 cpw 4533   cuni 4839  dom cdm 5589  cfv 6433  topGenctg 17148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-topgen 17154
This theorem is referenced by:  tgval3  22113  tgtop  22123  eltop3  22126  tgidm  22130  bastop1  22143  tgrest  22310  tgcn  22403  txbasval  22757  opnmblALT  24767  mbfimaopnlem  24819  isfne3  34532  fneuni  34536  dissneqlem  35511  tgqioo2  43085
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