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Theorem isbasis2g 22451
Description: Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
isbasis2g (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
Distinct variable group:   𝑥,𝑤,𝑦,𝑧,𝐵
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem isbasis2g
StepHypRef Expression
1 isbasisg 22450 . 2 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
2 dfss3 3971 . . . 4 ((𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑧 ∈ (𝑥𝑦)𝑧 (𝐵 ∩ 𝒫 (𝑥𝑦)))
3 elin 3965 . . . . . . . . . 10 (𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (𝑤𝐵𝑤 ∈ 𝒫 (𝑥𝑦)))
4 velpw 4608 . . . . . . . . . . 11 (𝑤 ∈ 𝒫 (𝑥𝑦) ↔ 𝑤 ⊆ (𝑥𝑦))
54anbi2i 624 . . . . . . . . . 10 ((𝑤𝐵𝑤 ∈ 𝒫 (𝑥𝑦)) ↔ (𝑤𝐵𝑤 ⊆ (𝑥𝑦)))
63, 5bitri 275 . . . . . . . . 9 (𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (𝑤𝐵𝑤 ⊆ (𝑥𝑦)))
76anbi2i 624 . . . . . . . 8 ((𝑧𝑤𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦))) ↔ (𝑧𝑤 ∧ (𝑤𝐵𝑤 ⊆ (𝑥𝑦))))
8 an12 644 . . . . . . . 8 ((𝑧𝑤 ∧ (𝑤𝐵𝑤 ⊆ (𝑥𝑦))) ↔ (𝑤𝐵 ∧ (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
97, 8bitri 275 . . . . . . 7 ((𝑧𝑤𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦))) ↔ (𝑤𝐵 ∧ (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
109exbii 1851 . . . . . 6 (∃𝑤(𝑧𝑤𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦))) ↔ ∃𝑤(𝑤𝐵 ∧ (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
11 eluni 4912 . . . . . 6 (𝑧 (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∃𝑤(𝑧𝑤𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦))))
12 df-rex 3072 . . . . . 6 (∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)) ↔ ∃𝑤(𝑤𝐵 ∧ (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1310, 11, 123bitr4i 303 . . . . 5 (𝑧 (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
1413ralbii 3094 . . . 4 (∀𝑧 ∈ (𝑥𝑦)𝑧 (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
152, 14bitri 275 . . 3 ((𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
16152ralbii 3129 . 2 (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
171, 16bitrdi 287 1 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wex 1782  wcel 2107  wral 3062  wrex 3071  cin 3948  wss 3949  𝒫 cpw 4603   cuni 4909  TopBasesctb 22448
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-pw 4605  df-uni 4910  df-bases 22449
This theorem is referenced by:  isbasis3g  22452  basis2  22454  fiinbas  22455  tgclb  22473  topbas  22475  restbas  22662  txbas  23071  blbas  23936
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