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Theorem isbasisg 12225
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 3270 . . . . . 6 (𝑧 = 𝐵 → (𝑧 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝑥𝑦)))
21unieqd 3747 . . . . 5 (𝑧 = 𝐵 (𝑧 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝑥𝑦)))
32sseq2d 3127 . . . 4 (𝑧 = 𝐵 → ((𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦)) ↔ (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
43raleqbi1dv 2634 . . 3 (𝑧 = 𝐵 → (∀𝑦𝑧 (𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
54raleqbi1dv 2634 . 2 (𝑧 = 𝐵 → (∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
6 df-bases 12224 . 2 TopBases = {𝑧 ∣ ∀𝑥𝑧𝑦𝑧 (𝑥𝑦) ⊆ (𝑧 ∩ 𝒫 (𝑥𝑦))}
75, 6elab2g 2831 1 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wcel 1480  wral 2416  cin 3070  wss 3071  𝒫 cpw 3510   cuni 3736  TopBasesctb 12223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-bases 12224
This theorem is referenced by:  isbasis2g  12226  basis1  12228  baspartn  12231
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