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Theorem isbasis2g 11994
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 11993 . 2 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
2 dfss3 3037 . . . 4 ((𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑧 ∈ (𝑥𝑦)𝑧 (𝐵 ∩ 𝒫 (𝑥𝑦)))
3 elin 3206 . . . . . . . . . 10 (𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (𝑤𝐵𝑤 ∈ 𝒫 (𝑥𝑦)))
4 selpw 3464 . . . . . . . . . . 11 (𝑤 ∈ 𝒫 (𝑥𝑦) ↔ 𝑤 ⊆ (𝑥𝑦))
54anbi2i 448 . . . . . . . . . 10 ((𝑤𝐵𝑤 ∈ 𝒫 (𝑥𝑦)) ↔ (𝑤𝐵𝑤 ⊆ (𝑥𝑦)))
63, 5bitri 183 . . . . . . . . 9 (𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (𝑤𝐵𝑤 ⊆ (𝑥𝑦)))
76anbi2i 448 . . . . . . . 8 ((𝑧𝑤𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦))) ↔ (𝑧𝑤 ∧ (𝑤𝐵𝑤 ⊆ (𝑥𝑦))))
8 an12 531 . . . . . . . 8 ((𝑧𝑤 ∧ (𝑤𝐵𝑤 ⊆ (𝑥𝑦))) ↔ (𝑤𝐵 ∧ (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
97, 8bitri 183 . . . . . . 7 ((𝑧𝑤𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦))) ↔ (𝑤𝐵 ∧ (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
109exbii 1552 . . . . . 6 (∃𝑤(𝑧𝑤𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦))) ↔ ∃𝑤(𝑤𝐵 ∧ (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
11 eluni 3686 . . . . . 6 (𝑧 (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∃𝑤(𝑧𝑤𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦))))
12 df-rex 2381 . . . . . 6 (∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)) ↔ ∃𝑤(𝑤𝐵 ∧ (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1310, 11, 123bitr4i 211 . . . . 5 (𝑧 (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
1413ralbii 2400 . . . 4 (∀𝑧 ∈ (𝑥𝑦)𝑧 (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
152, 14bitri 183 . . 3 ((𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
16152ralbii 2402 . 2 (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
171, 16syl6bb 195 1 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wex 1436  wcel 1448  wral 2375  wrex 2376  cin 3020  wss 3021  𝒫 cpw 3457   cuni 3683  TopBasesctb 11991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-in 3027  df-ss 3034  df-pw 3459  df-uni 3684  df-bases 11992
This theorem is referenced by:  isbasis3g  11995  basis2  11997  fiinbas  11998  tgclb  12016  topbas  12018  restbasg  12119  txbas  12208  blbas  12361
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