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Theorem isbasis2g 21532
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 21531 . 2 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
2 dfss3 3932 . . . 4 ((𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑧 ∈ (𝑥𝑦)𝑧 (𝐵 ∩ 𝒫 (𝑥𝑦)))
3 elin 3926 . . . . . . . . . 10 (𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (𝑤𝐵𝑤 ∈ 𝒫 (𝑥𝑦)))
4 velpw 4517 . . . . . . . . . . 11 (𝑤 ∈ 𝒫 (𝑥𝑦) ↔ 𝑤 ⊆ (𝑥𝑦))
54anbi2i 625 . . . . . . . . . 10 ((𝑤𝐵𝑤 ∈ 𝒫 (𝑥𝑦)) ↔ (𝑤𝐵𝑤 ⊆ (𝑥𝑦)))
63, 5bitri 278 . . . . . . . . 9 (𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (𝑤𝐵𝑤 ⊆ (𝑥𝑦)))
76anbi2i 625 . . . . . . . 8 ((𝑧𝑤𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦))) ↔ (𝑧𝑤 ∧ (𝑤𝐵𝑤 ⊆ (𝑥𝑦))))
8 an12 644 . . . . . . . 8 ((𝑧𝑤 ∧ (𝑤𝐵𝑤 ⊆ (𝑥𝑦))) ↔ (𝑤𝐵 ∧ (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
97, 8bitri 278 . . . . . . 7 ((𝑧𝑤𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦))) ↔ (𝑤𝐵 ∧ (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
109exbii 1849 . . . . . 6 (∃𝑤(𝑧𝑤𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦))) ↔ ∃𝑤(𝑤𝐵 ∧ (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
11 eluni 4814 . . . . . 6 (𝑧 (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∃𝑤(𝑧𝑤𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦))))
12 df-rex 3132 . . . . . 6 (∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)) ↔ ∃𝑤(𝑤𝐵 ∧ (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1310, 11, 123bitr4i 306 . . . . 5 (𝑧 (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
1413ralbii 3153 . . . 4 (∀𝑧 ∈ (𝑥𝑦)𝑧 (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
152, 14bitri 278 . . 3 ((𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
16152ralbii 3154 . 2 (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
171, 16syl6bb 290 1 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wex 1781  wcel 2115  wral 3126  wrex 3127  cin 3909  wss 3910  𝒫 cpw 4512   cuni 4811  TopBasesctb 21529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-in 3917  df-ss 3927  df-pw 4514  df-uni 4812  df-bases 21530
This theorem is referenced by:  isbasis3g  21533  basis2  21535  fiinbas  21536  tgclb  21554  topbas  21556  restbas  21742  txbas  22151  blbas  23016
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