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Theorem eltg3 21486
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 6699 . . . 4 (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen)
2 inex1g 5220 . . . 4 (𝐵 ∈ dom topGen → (𝐵 ∩ 𝒫 𝐴) ∈ V)
31, 2syl 17 . . 3 (𝐴 ∈ (topGen‘𝐵) → (𝐵 ∩ 𝒫 𝐴) ∈ V)
4 eltg4i 21484 . . 3 (𝐴 ∈ (topGen‘𝐵) → 𝐴 = (𝐵 ∩ 𝒫 𝐴))
5 inss1 4209 . . . . . 6 (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐵
6 sseq1 3996 . . . . . 6 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → (𝑥𝐵 ↔ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐵))
75, 6mpbiri 259 . . . . 5 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → 𝑥𝐵)
87biantrurd 533 . . . 4 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → (𝐴 = 𝑥 ↔ (𝑥𝐵𝐴 = 𝑥)))
9 unieq 4845 . . . . 5 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → 𝑥 = (𝐵 ∩ 𝒫 𝐴))
109eqeq2d 2837 . . . 4 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → (𝐴 = 𝑥𝐴 = (𝐵 ∩ 𝒫 𝐴)))
118, 10bitr3d 282 . . 3 (𝑥 = (𝐵 ∩ 𝒫 𝐴) → ((𝑥𝐵𝐴 = 𝑥) ↔ 𝐴 = (𝐵 ∩ 𝒫 𝐴)))
123, 4, 11elabd 3672 . 2 (𝐴 ∈ (topGen‘𝐵) → ∃𝑥(𝑥𝐵𝐴 = 𝑥))
13 eltg3i 21485 . . . . 5 ((𝐵𝑉𝑥𝐵) → 𝑥 ∈ (topGen‘𝐵))
14 eleq1 2905 . . . . 5 (𝐴 = 𝑥 → (𝐴 ∈ (topGen‘𝐵) ↔ 𝑥 ∈ (topGen‘𝐵)))
1513, 14syl5ibrcom 248 . . . 4 ((𝐵𝑉𝑥𝐵) → (𝐴 = 𝑥𝐴 ∈ (topGen‘𝐵)))
1615expimpd 454 . . 3 (𝐵𝑉 → ((𝑥𝐵𝐴 = 𝑥) → 𝐴 ∈ (topGen‘𝐵)))
1716exlimdv 1927 . 2 (𝐵𝑉 → (∃𝑥(𝑥𝐵𝐴 = 𝑥) → 𝐴 ∈ (topGen‘𝐵)))
1812, 17impbid2 227 1 (𝐵𝑉 → (𝐴 ∈ (topGen‘𝐵) ↔ ∃𝑥(𝑥𝐵𝐴 = 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  Vcvv 3500  cin 3939  wss 3940  𝒫 cpw 4542   cuni 4837  dom cdm 5554  cfv 6352  topGenctg 16701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6312  df-fun 6354  df-fv 6360  df-topgen 16707
This theorem is referenced by:  tgval3  21487  tgtop  21497  eltop3  21500  tgidm  21504  bastop1  21517  tgrest  21683  tgcn  21776  txbasval  22130  opnmblALT  24119  mbfimaopnlem  24171  isfne3  33575  fneuni  33579  dissneqlem  34490  tgqioo2  41688
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