MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-bases Structured version   Visualization version   GIF version

Definition df-bases 20469
Description: Define the class of all topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 20510). Note that "bases" is the plural of "basis." (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
df-bases TopBases = {𝑥 ∣ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ⊆ (𝑥 ∩ 𝒫 (𝑦𝑧))}
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-bases
StepHypRef Expression
1 ctb 20467 . 2 class TopBases
2 vy . . . . . . . 8 setvar 𝑦
32cv 1473 . . . . . . 7 class 𝑦
4 vz . . . . . . . 8 setvar 𝑧
54cv 1473 . . . . . . 7 class 𝑧
63, 5cin 3538 . . . . . 6 class (𝑦𝑧)
7 vx . . . . . . . . 9 setvar 𝑥
87cv 1473 . . . . . . . 8 class 𝑥
96cpw 4107 . . . . . . . 8 class 𝒫 (𝑦𝑧)
108, 9cin 3538 . . . . . . 7 class (𝑥 ∩ 𝒫 (𝑦𝑧))
1110cuni 4366 . . . . . 6 class (𝑥 ∩ 𝒫 (𝑦𝑧))
126, 11wss 3539 . . . . 5 wff (𝑦𝑧) ⊆ (𝑥 ∩ 𝒫 (𝑦𝑧))
1312, 4, 8wral 2895 . . . 4 wff 𝑧𝑥 (𝑦𝑧) ⊆ (𝑥 ∩ 𝒫 (𝑦𝑧))
1413, 2, 8wral 2895 . . 3 wff 𝑦𝑥𝑧𝑥 (𝑦𝑧) ⊆ (𝑥 ∩ 𝒫 (𝑦𝑧))
1514, 7cab 2595 . 2 class {𝑥 ∣ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ⊆ (𝑥 ∩ 𝒫 (𝑦𝑧))}
161, 15wceq 1474 1 wff TopBases = {𝑥 ∣ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ⊆ (𝑥 ∩ 𝒫 (𝑦𝑧))}
Colors of variables: wff setvar class
This definition is referenced by:  isbasisg  20509
  Copyright terms: Public domain W3C validator