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Theorem isbasisg 21550
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 4155 . . . . . 6 (𝑧 = 𝐵 → (𝑧 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝑥𝑦)))
21unieqd 4827 . . . . 5 (𝑧 = 𝐵 (𝑧 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝑥𝑦)))
32sseq2d 3974 . . . 4 (𝑧 = 𝐵 → ((𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦)) ↔ (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
43raleqbi1dv 3384 . . 3 (𝑧 = 𝐵 → (∀𝑦𝑧 (𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
54raleqbi1dv 3384 . 2 (𝑧 = 𝐵 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
6 df-bases 21549 . 2 TopBases = {𝑧 ∣ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦))}
75, 6elab2g 3643 1 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2114  wral 3130  cin 3907  wss 3908  𝒫 cpw 4511   cuni 4813  TopBasesctb 21548
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
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 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rab 3139  df-v 3471  df-in 3915  df-ss 3925  df-uni 4814  df-bases 21549
This theorem is referenced by:  isbasis2g  21551  basis1  21553  basdif0  21556  baspartn  21557  basqtop  22314
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