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

Definition df-lly 21017
Description: Define a space that is locally 𝐴, where 𝐴 is a topological property like "compact", "connected", or "path-connected". A topological space is locally 𝐴 if every neighborhood of a point contains an open sub-neighborhood that is 𝐴 in the subspace topology. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
df-lly Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)}
Distinct variable group:   𝑢,𝑗,𝑥,𝑦,𝐴

Detailed syntax breakdown of Definition df-lly
StepHypRef Expression
1 cA . . 3 class 𝐴
21clly 21015 . 2 class Locally 𝐴
3 vy . . . . . . . 8 setvar 𝑦
4 vu . . . . . . . 8 setvar 𝑢
53, 4wel 1976 . . . . . . 7 wff 𝑦𝑢
6 vj . . . . . . . . . 10 setvar 𝑗
76cv 1473 . . . . . . . . 9 class 𝑗
84cv 1473 . . . . . . . . 9 class 𝑢
9 crest 15846 . . . . . . . . 9 class t
107, 8, 9co 6523 . . . . . . . 8 class (𝑗t 𝑢)
1110, 1wcel 1975 . . . . . . 7 wff (𝑗t 𝑢) ∈ 𝐴
125, 11wa 382 . . . . . 6 wff (𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)
13 vx . . . . . . . . 9 setvar 𝑥
1413cv 1473 . . . . . . . 8 class 𝑥
1514cpw 4103 . . . . . . 7 class 𝒫 𝑥
167, 15cin 3534 . . . . . 6 class (𝑗 ∩ 𝒫 𝑥)
1712, 4, 16wrex 2892 . . . . 5 wff 𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)
1817, 3, 14wral 2891 . . . 4 wff 𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)
1918, 13, 7wral 2891 . . 3 wff 𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)
20 ctop 20455 . . 3 class Top
2119, 6, 20crab 2895 . 2 class {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)}
222, 21wceq 1474 1 wff Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)}
Colors of variables: wff setvar class
This definition is referenced by:  islly  21019  llyeq  21021
  Copyright terms: Public domain W3C validator