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Theorem 1stcclb 17507
Description: A property of points in a first-countable topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
Hypothesis
Ref Expression
1stcclb.1  |-  X  = 
U. J
Assertion
Ref Expression
1stcclb  |-  ( ( J  e.  1stc  /\  A  e.  X )  ->  E. x  e.  ~P  J ( x  ~<_  om  /\  A. y  e.  J  ( A  e.  y  ->  E. z  e.  x  ( A  e.  z  /\  z  C_  y ) ) ) )
Distinct variable groups:    x, y,
z, A    x, J, y, z    x, X, y, z

Proof of Theorem 1stcclb
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 1stcclb.1 . . . 4  |-  X  = 
U. J
21is1stc2 17505 . . 3  |-  ( J  e.  1stc  <->  ( J  e. 
Top  /\  A. w  e.  X  E. x  e.  ~P  J ( x  ~<_  om  /\  A. y  e.  J  ( w  e.  y  ->  E. z  e.  x  ( w  e.  z  /\  z  C_  y ) ) ) ) )
32simprbi 451 . 2  |-  ( J  e.  1stc  ->  A. w  e.  X  E. x  e.  ~P  J ( x  ~<_  om  /\  A. y  e.  J  ( w  e.  y  ->  E. z  e.  x  ( w  e.  z  /\  z  C_  y ) ) ) )
4 eleq1 2496 . . . . . . 7  |-  ( w  =  A  ->  (
w  e.  y  <->  A  e.  y ) )
5 eleq1 2496 . . . . . . . . 9  |-  ( w  =  A  ->  (
w  e.  z  <->  A  e.  z ) )
65anbi1d 686 . . . . . . . 8  |-  ( w  =  A  ->  (
( w  e.  z  /\  z  C_  y
)  <->  ( A  e.  z  /\  z  C_  y ) ) )
76rexbidv 2726 . . . . . . 7  |-  ( w  =  A  ->  ( E. z  e.  x  ( w  e.  z  /\  z  C_  y )  <->  E. z  e.  x  ( A  e.  z  /\  z  C_  y ) ) )
84, 7imbi12d 312 . . . . . 6  |-  ( w  =  A  ->  (
( w  e.  y  ->  E. z  e.  x  ( w  e.  z  /\  z  C_  y ) )  <->  ( A  e.  y  ->  E. z  e.  x  ( A  e.  z  /\  z  C_  y ) ) ) )
98ralbidv 2725 . . . . 5  |-  ( w  =  A  ->  ( A. y  e.  J  ( w  e.  y  ->  E. z  e.  x  ( w  e.  z  /\  z  C_  y ) )  <->  A. y  e.  J  ( A  e.  y  ->  E. z  e.  x  ( A  e.  z  /\  z  C_  y ) ) ) )
109anbi2d 685 . . . 4  |-  ( w  =  A  ->  (
( x  ~<_  om  /\  A. y  e.  J  ( w  e.  y  ->  E. z  e.  x  ( w  e.  z  /\  z  C_  y ) ) )  <->  ( x  ~<_  om  /\  A. y  e.  J  ( A  e.  y  ->  E. z  e.  x  ( A  e.  z  /\  z  C_  y ) ) ) ) )
1110rexbidv 2726 . . 3  |-  ( w  =  A  ->  ( E. x  e.  ~P  J ( x  ~<_  om 
/\  A. y  e.  J  ( w  e.  y  ->  E. z  e.  x  ( w  e.  z  /\  z  C_  y ) ) )  <->  E. x  e.  ~P  J ( x  ~<_  om  /\  A. y  e.  J  ( A  e.  y  ->  E. z  e.  x  ( A  e.  z  /\  z  C_  y ) ) ) ) )
1211rspcv 3048 . 2  |-  ( A  e.  X  ->  ( A. w  e.  X  E. x  e.  ~P  J ( x  ~<_  om 
/\  A. y  e.  J  ( w  e.  y  ->  E. z  e.  x  ( w  e.  z  /\  z  C_  y ) ) )  ->  E. x  e.  ~P  J ( x  ~<_  om  /\  A. y  e.  J  ( A  e.  y  ->  E. z  e.  x  ( A  e.  z  /\  z  C_  y ) ) ) ) )
133, 12mpan9 456 1  |-  ( ( J  e.  1stc  /\  A  e.  X )  ->  E. x  e.  ~P  J ( x  ~<_  om  /\  A. y  e.  J  ( A  e.  y  ->  E. z  e.  x  ( A  e.  z  /\  z  C_  y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    C_ wss 3320   ~Pcpw 3799   U.cuni 4015   class class class wbr 4212   omcom 4845    ~<_ cdom 7107   Topctop 16958   1stcc1stc 17500
This theorem is referenced by:  1stcfb  17508  1stcrest  17516  lly1stc  17559  tx1stc  17682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-in 3327  df-ss 3334  df-pw 3801  df-uni 4016  df-1stc 17502
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