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Theorem isbasisg 22969
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 4220 . . . . . 6 (𝑧 = 𝐵 → (𝑧 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝑥𝑦)))
21unieqd 4924 . . . . 5 (𝑧 = 𝐵 (𝑧 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝑥𝑦)))
32sseq2d 4027 . . . 4 (𝑧 = 𝐵 → ((𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦)) ↔ (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
43raleqbi1dv 3335 . . 3 (𝑧 = 𝐵 → (∀𝑦𝑧 (𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
54raleqbi1dv 3335 . 2 (𝑧 = 𝐵 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
6 df-bases 22968 . 2 TopBases = {𝑧 ∣ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦))}
75, 6elab2g 3682 1 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1536  wcel 2105  wral 3058  cin 3961  wss 3962  𝒫 cpw 4604   cuni 4911  TopBasesctb 22967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-in 3969  df-ss 3979  df-uni 4912  df-bases 22968
This theorem is referenced by:  isbasis2g  22970  basis1  22972  basdif0  22975  baspartn  22976  basqtop  23734
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