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Theorem isbasisg 22938
Description: Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
isbasisg (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
Distinct variable group:   𝑥,𝑦,𝐵
Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem isbasisg
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ineq1 4203 . . . . . 6 (𝑧 = 𝐵 → (𝑧 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝑥𝑦)))
21unieqd 4918 . . . . 5 (𝑧 = 𝐵 (𝑧 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝑥𝑦)))
32sseq2d 4011 . . . 4 (𝑧 = 𝐵 → ((𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦)) ↔ (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
43raleqbi1dv 3323 . . 3 (𝑧 = 𝐵 → (∀𝑦𝑧 (𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
54raleqbi1dv 3323 . 2 (𝑧 = 𝐵 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
6 df-bases 22937 . 2 TopBases = {𝑧 ∣ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦))}
75, 6elab2g 3667 1 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  wral 3051  cin 3945  wss 3946  𝒫 cpw 4597   cuni 4905  TopBasesctb 22936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-in 3953  df-ss 3963  df-uni 4906  df-bases 22937
This theorem is referenced by:  isbasis2g  22939  basis1  22941  basdif0  22944  baspartn  22945  basqtop  23703
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