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

Theorem llytop 19839
Description: A locally  A space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llytop  |-  ( J  e. Locally  A  ->  J  e. 
Top )

Proof of Theorem llytop
Dummy variables  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 islly 19835 . 2  |-  ( J  e. Locally  A  <->  ( J  e. 
Top  /\  A. x  e.  J  A. y  e.  x  E. u  e.  ( J  i^i  ~P x ) ( y  e.  u  /\  ( Jt  u )  e.  A
) ) )
21simplbi 460 1  |-  ( J  e. Locally  A  ->  J  e. 
Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   A.wral 2817   E.wrex 2818    i^i cin 3480   ~Pcpw 4016  (class class class)co 6295   ↾t crest 14692   Topctop 19261  Locally clly 19831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298  df-lly 19833
This theorem is referenced by:  llynlly  19844  islly2  19851  llyrest  19852  llyidm  19855  nllyidm  19856  toplly  19857  lly1stc  19863  txlly  20003
  Copyright terms: Public domain W3C validator