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Theorem isbasis2g 20563
 Description: Express the predicate "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 20562 . 2 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
2 dfss3 3558 . . . 4 ((𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑧 ∈ (𝑥𝑦)𝑧 (𝐵 ∩ 𝒫 (𝑥𝑦)))
3 elin 3758 . . . . . . . . . 10 (𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (𝑤𝐵𝑤 ∈ 𝒫 (𝑥𝑦)))
4 selpw 4115 . . . . . . . . . . 11 (𝑤 ∈ 𝒫 (𝑥𝑦) ↔ 𝑤 ⊆ (𝑥𝑦))
54anbi2i 726 . . . . . . . . . 10 ((𝑤𝐵𝑤 ∈ 𝒫 (𝑥𝑦)) ↔ (𝑤𝐵𝑤 ⊆ (𝑥𝑦)))
63, 5bitri 263 . . . . . . . . 9 (𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (𝑤𝐵𝑤 ⊆ (𝑥𝑦)))
76anbi2i 726 . . . . . . . 8 ((𝑧𝑤𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦))) ↔ (𝑧𝑤 ∧ (𝑤𝐵𝑤 ⊆ (𝑥𝑦))))
8 an12 834 . . . . . . . 8 ((𝑧𝑤 ∧ (𝑤𝐵𝑤 ⊆ (𝑥𝑦))) ↔ (𝑤𝐵 ∧ (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
97, 8bitri 263 . . . . . . 7 ((𝑧𝑤𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦))) ↔ (𝑤𝐵 ∧ (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
109exbii 1764 . . . . . 6 (∃𝑤(𝑧𝑤𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦))) ↔ ∃𝑤(𝑤𝐵 ∧ (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
11 eluni 4375 . . . . . 6 (𝑧 (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∃𝑤(𝑧𝑤𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥𝑦))))
12 df-rex 2902 . . . . . 6 (∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)) ↔ ∃𝑤(𝑤𝐵 ∧ (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1310, 11, 123bitr4i 291 . . . . 5 (𝑧 (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
1413ralbii 2963 . . . 4 (∀𝑧 ∈ (𝑥𝑦)𝑧 (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
152, 14bitri 263 . . 3 ((𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
16152ralbii 2964 . 2 (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
171, 16syl6bb 275 1 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  TopBasesctb 20520 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373  df-bases 20522 This theorem is referenced by:  isbasis3g  20564  basis2  20566  fiinbas  20567  tgclb  20585  topbas  20587  restbas  20772  txbas  21180  blbas  22045
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