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

Theorem islly 21081
 Description: The property of being a locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
islly (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
Distinct variable groups:   𝑥,𝑢,𝑦,𝐴   𝑢,𝐽,𝑥,𝑦

Proof of Theorem islly
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 ineq1 3769 . . . . 5 (𝑗 = 𝐽 → (𝑗 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑥))
2 oveq1 6556 . . . . . . 7 (𝑗 = 𝐽 → (𝑗t 𝑢) = (𝐽t 𝑢))
32eleq1d 2672 . . . . . 6 (𝑗 = 𝐽 → ((𝑗t 𝑢) ∈ 𝐴 ↔ (𝐽t 𝑢) ∈ 𝐴))
43anbi2d 736 . . . . 5 (𝑗 = 𝐽 → ((𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) ↔ (𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
51, 4rexeqbidv 3130 . . . 4 (𝑗 = 𝐽 → (∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) ↔ ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
65ralbidv 2969 . . 3 (𝑗 = 𝐽 → (∀𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) ↔ ∀𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
76raleqbi1dv 3123 . 2 (𝑗 = 𝐽 → (∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴) ↔ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
8 df-lly 21079 . 2 Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝑗t 𝑢) ∈ 𝐴)}
97, 8elrab2 3333 1 (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539  𝒫 cpw 4108  (class class class)co 6549   ↾t crest 15904  Topctop 20517  Locally clly 21077 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-lly 21079 This theorem is referenced by:  llytop  21085  llyi  21087  llyss  21092  subislly  21094  restnlly  21095  restlly  21096  islly2  21097  llyrest  21098  llyidm  21101  dislly  21110  txlly  21249  ismntop  29398  cnllyscon  30481  rellyscon  30487
 Copyright terms: Public domain W3C validator