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Theorem is1stc 19708
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 4253 . . . 4  |-  ( j  =  J  ->  U. j  =  U. J )
2 is1stc.1 . . . 4  |-  X  = 
U. J
31, 2syl6eqr 2526 . . 3  |-  ( j  =  J  ->  U. j  =  X )
4 pweq 4013 . . . 4  |-  ( j  =  J  ->  ~P j  =  ~P J
)
5 raleq 3058 . . . . 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 703 . . . 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 3073 . . 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 3072 . 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 19706 . 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 3263 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    i^i cin 3475   ~Pcpw 4010   U.cuni 4245   class class class wbr 4447   omcom 6678    ~<_ cdom 7511   Topctop 19161   1stcc1stc 19704
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-an 371  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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-in 3483  df-ss 3490  df-pw 4012  df-uni 4246  df-1stc 19706
This theorem is referenced by:  is1stc2  19709  1stctop  19710
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