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Theorem 1stcclb 19053
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 19051 . . 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 464 . 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 2503 . . . . . . 7  |-  ( w  =  A  ->  (
w  e.  y  <->  A  e.  y ) )
5 eleq1 2503 . . . . . . . . 9  |-  ( w  =  A  ->  (
w  e.  z  <->  A  e.  z ) )
65anbi1d 704 . . . . . . . 8  |-  ( w  =  A  ->  (
( w  e.  z  /\  z  C_  y
)  <->  ( A  e.  z  /\  z  C_  y ) ) )
76rexbidv 2741 . . . . . . 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 320 . . . . . 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 2740 . . . . 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 703 . . . 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 2741 . . 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 3074 . 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 469 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721    C_ wss 3333   ~Pcpw 3865   U.cuni 4096   class class class wbr 4297   omcom 6481    ~<_ cdom 7313   Topctop 18503   1stcc1stc 19046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-in 3340  df-ss 3347  df-pw 3867  df-uni 4097  df-1stc 19048
This theorem is referenced by:  1stcfb  19054  1stcrest  19062  lly1stc  19105  tx1stc  19228
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