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Theorem 1stcclb 17170
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 17168 . . 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 450 . 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 2343 . . . . . . 7  |-  ( w  =  A  ->  (
w  e.  y  <->  A  e.  y ) )
5 eleq1 2343 . . . . . . . . 9  |-  ( w  =  A  ->  (
w  e.  z  <->  A  e.  z ) )
65anbi1d 685 . . . . . . . 8  |-  ( w  =  A  ->  (
( w  e.  z  /\  z  C_  y
)  <->  ( A  e.  z  /\  z  C_  y ) ) )
76rexbidv 2564 . . . . . . 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 311 . . . . . 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 2563 . . . . 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 684 . . . 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 2564 . . 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 2880 . 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 455 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 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   class class class wbr 4023   omcom 4656    ~<_ cdom 6861   Topctop 16631   1stcc1stc 17163
This theorem is referenced by:  1stcfb  17171  1stcrest  17179  lly1stc  17222  tx1stc  17344
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
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