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Theorem isbasisg 11993
 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 3217 . . . . . 6 (𝑧 = 𝐵 → (𝑧 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝑥𝑦)))
21unieqd 3694 . . . . 5 (𝑧 = 𝐵 (𝑧 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝑥𝑦)))
32sseq2d 3077 . . . 4 (𝑧 = 𝐵 → ((𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦)) ↔ (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
43raleqbi1dv 2592 . . 3 (𝑧 = 𝐵 → (∀𝑦𝑧 (𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
54raleqbi1dv 2592 . 2 (𝑧 = 𝐵 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
6 df-bases 11992 . 2 TopBases = {𝑧 ∣ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦))}
75, 6elab2g 2784 1 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1299   ∈ wcel 1448  ∀wral 2375   ∩ cin 3020   ⊆ wss 3021  𝒫 cpw 3457  ∪ cuni 3683  TopBasesctb 11991 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082 This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-in 3027  df-ss 3034  df-uni 3684  df-bases 11992 This theorem is referenced by:  isbasis2g  11994  basis1  11996  baspartn  11999
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