MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  is1stc Unicode version

Theorem is1stc 17167
Description: The predicate "is a first-countable topology." This can be described as "every point has a countable local basis" - that is, every point has a countable collection of open sets containing it such that every open set containing the point has an open set from this collection as a subset. (Contributed by Jeff Hankins, 22-Aug-2009.)
Hypothesis
Ref Expression
is1stc.1  |-  X  = 
U. J
Assertion
Ref Expression
is1stc  |-  ( J  e.  1stc  <->  ( J  e. 
Top  /\  A. x  e.  X  E. y  e.  ~P  J ( y  ~<_  om  /\  A. z  e.  J  ( x  e.  z  ->  x  e. 
U. ( y  i^i 
~P z ) ) ) ) )
Distinct variable groups:    x, y,
z, J    x, X
Allowed substitution hints:    X( y, z)

Proof of Theorem is1stc
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 unieq 3836 . . . 4  |-  ( j  =  J  ->  U. j  =  U. J )
2 is1stc.1 . . . 4  |-  X  = 
U. J
31, 2syl6eqr 2333 . . 3  |-  ( j  =  J  ->  U. j  =  X )
4 pweq 3628 . . . 4  |-  ( j  =  J  ->  ~P j  =  ~P J
)
5 raleq 2736 . . . . 5  |-  ( j  =  J  ->  ( A. z  e.  j 
( x  e.  z  ->  x  e.  U. ( y  i^i  ~P z ) )  <->  A. z  e.  J  ( x  e.  z  ->  x  e. 
U. ( y  i^i 
~P z ) ) ) )
65anbi2d 684 . . . 4  |-  ( j  =  J  ->  (
( y  ~<_  om  /\  A. z  e.  j  ( x  e.  z  ->  x  e.  U. (
y  i^i  ~P z
) ) )  <->  ( y  ~<_  om  /\  A. z  e.  J  ( x  e.  z  ->  x  e.  U. ( y  i^i  ~P z ) ) ) ) )
74, 6rexeqbidv 2749 . . 3  |-  ( j  =  J  ->  ( E. y  e.  ~P  j ( y  ~<_  om 
/\  A. z  e.  j  ( x  e.  z  ->  x  e.  U. ( y  i^i  ~P z ) ) )  <->  E. y  e.  ~P  J ( y  ~<_  om 
/\  A. z  e.  J  ( x  e.  z  ->  x  e.  U. (
y  i^i  ~P z
) ) ) ) )
83, 7raleqbidv 2748 . 2  |-  ( j  =  J  ->  ( A. x  e.  U. j E. y  e.  ~P  j ( y  ~<_  om 
/\  A. z  e.  j  ( x  e.  z  ->  x  e.  U. ( y  i^i  ~P z ) ) )  <->  A. x  e.  X  E. y  e.  ~P  J ( y  ~<_  om 
/\  A. z  e.  J  ( x  e.  z  ->  x  e.  U. (
y  i^i  ~P z
) ) ) ) )
9 df-1stc 17165 . 2  |-  1stc  =  { j  e.  Top  | 
A. x  e.  U. j E. y  e.  ~P  j ( y  ~<_  om 
/\  A. z  e.  j  ( x  e.  z  ->  x  e.  U. ( y  i^i  ~P z ) ) ) }
108, 9elrab2 2925 1  |-  ( J  e.  1stc  <->  ( J  e. 
Top  /\  A. x  e.  X  E. y  e.  ~P  J ( y  ~<_  om  /\  A. z  e.  J  ( x  e.  z  ->  x  e. 
U. ( y  i^i 
~P z ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151   ~Pcpw 3625   U.cuni 3827   class class class wbr 4023   omcom 4656    ~<_ cdom 6861   Topctop 16631   1stcc1stc 17163
This theorem is referenced by:  is1stc2  17168  1stctop  17169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-uni 3828  df-1stc 17165
  Copyright terms: Public domain W3C validator