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

Theorem llytop 22823
Description: A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llytop (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)

Proof of Theorem llytop
Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 islly 22819 . 2 (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)))
21simplbi 498 1 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wral 3064  wrex 3073  cin 3909  𝒫 cpw 4560  (class class class)co 7357  t crest 17302  Topctop 22242  Locally clly 22815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7360  df-lly 22817
This theorem is referenced by:  llynlly  22828  islly2  22835  llyrest  22836  llyidm  22839  nllyidm  22840  toplly  22841  lly1stc  22847  txlly  22987
  Copyright terms: Public domain W3C validator