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Theorem is1stc 17427
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 3968 . . . 4  |-  ( j  =  J  ->  U. j  =  U. J )
2 is1stc.1 . . . 4  |-  X  = 
U. J
31, 2syl6eqr 2439 . . 3  |-  ( j  =  J  ->  U. j  =  X )
4 pweq 3747 . . . 4  |-  ( j  =  J  ->  ~P j  =  ~P J
)
5 raleq 2849 . . . . 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 685 . . . 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 2862 . . 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 2861 . 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 17425 . 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 3039 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   E.wrex 2652    i^i cin 3264   ~Pcpw 3744   U.cuni 3959   class class class wbr 4155   omcom 4787    ~<_ cdom 7045   Topctop 16883   1stcc1stc 17423
This theorem is referenced by:  is1stc2  17428  1stctop  17429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-in 3272  df-ss 3279  df-pw 3746  df-uni 3960  df-1stc 17425
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