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Theorem eltg3 22969
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 6943 . . . 4 (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen)
2 inex1g 5319 . . . 4 (𝐵 ∈ dom topGen → (𝐵 ∩ 𝒫 𝐴) ∈ V)
31, 2syl 17 . . 3 (𝐴 ∈ (topGen‘𝐵) → (𝐵 ∩ 𝒫 𝐴) ∈ V)
4 eltg4i 22967 . . 3 (𝐴 ∈ (topGen‘𝐵) → 𝐴 = (𝐵 ∩ 𝒫 𝐴))
5 inss1 4237 . . . . . 6 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐵
6 sseq1 4009 . . . . . 6 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → (𝑥𝐵 ↔ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐵))
75, 6mpbiri 258 . . . . 5 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → 𝑥𝐵)
87biantrurd 532 . . . 4 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → (𝐴 = 𝑥 ↔ (𝑥𝐵𝐴 = 𝑥)))
9 unieq 4918 . . . . 5 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → 𝑥 = (𝐵 ∩ 𝒫 𝐴))
109eqeq2d 2748 . . . 4 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → (𝐴 = 𝑥𝐴 = (𝐵 ∩ 𝒫 𝐴)))
118, 10bitr3d 281 . . 3 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → ((𝑥𝐵𝐴 = 𝑥) ↔ 𝐴 = (𝐵 ∩ 𝒫 𝐴)))
123, 4, 11spcedv 3598 . 2 (𝐴 ∈ (topGen‘𝐵) → ∃𝑥(𝑥𝐵𝐴 = 𝑥))
13 eltg3i 22968 . . . . 5 ((𝐵𝑉𝑥𝐵) → 𝑥 ∈ (topGen‘𝐵))
14 eleq1 2829 . . . . 5 (𝐴 = 𝑥 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝑥 ∈ (topGen‘𝐵)))
1513, 14syl5ibrcom 247 . . . 4 ((𝐵𝑉𝑥𝐵) → (𝐴 = 𝑥𝐴 ∈ (topGen‘𝐵)))
1615expimpd 453 . . 3 (𝐵𝑉 → ((𝑥𝐵𝐴 = 𝑥) → 𝐴 ∈ (topGen‘𝐵)))
1716exlimdv 1933 . 2 (𝐵𝑉 → (∃𝑥(𝑥𝐵𝐴 = 𝑥) → 𝐴 ∈ (topGen‘𝐵)))
1812, 17impbid2 226 1 (𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥𝐵𝐴 = 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907  dom cdm 5685  cfv 6561  topGenctg 17482
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-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-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-topgen 17488
This theorem is referenced by:  tgval3  22970  tgtop  22980  eltop3  22983  tgidm  22987  bastop1  23000  tgrest  23167  tgcn  23260  txbasval  23614  opnmblALT  25638  mbfimaopnlem  25690  isfne3  36344  fneuni  36348  dissneqlem  37341  tgqioo2  45560
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