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Theorem 1stcclb 17186
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 17184 . . 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 2356 . . . . . . 7  |-  ( w  =  A  ->  (
w  e.  y  <->  A  e.  y ) )
5 eleq1 2356 . . . . . . . . 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 2577 . . . . . . 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 2576 . . . . 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 2577 . . 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 2893 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   class class class wbr 4039   omcom 4672    ~<_ cdom 6877   Topctop 16647   1stcc1stc 17179
This theorem is referenced by:  1stcfb  17187  1stcrest  17195  lly1stc  17238  tx1stc  17360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-in 3172  df-ss 3179  df-pw 3640  df-uni 3844  df-1stc 17181
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