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Theorem is2ndc 19181
Description: The property of being second-countable. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
is2ndc  |-  ( J  e.  2ndc  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) )
Distinct variable group:    x, J

Proof of Theorem is2ndc
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 df-2ndc 19175 . . 3  |-  2ndc  =  { j  |  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x
)  =  j ) }
21eleq2i 2532 . 2  |-  ( J  e.  2ndc  <->  J  e.  { j  |  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  j ) } )
3 simpr 461 . . . . 5  |-  ( ( x  ~<_  om  /\  ( topGen `
 x )  =  J )  ->  ( topGen `
 x )  =  J )
4 fvex 5808 . . . . 5  |-  ( topGen `  x )  e.  _V
53, 4syl6eqelr 2551 . . . 4  |-  ( ( x  ~<_  om  /\  ( topGen `
 x )  =  J )  ->  J  e.  _V )
65rexlimivw 2941 . . 3  |-  ( E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  J )  ->  J  e.  _V )
7 eqeq2 2469 . . . . 5  |-  ( j  =  J  ->  (
( topGen `  x )  =  j  <->  ( topGen `  x
)  =  J ) )
87anbi2d 703 . . . 4  |-  ( j  =  J  ->  (
( x  ~<_  om  /\  ( topGen `  x )  =  j )  <->  ( x  ~<_  om  /\  ( topGen `  x
)  =  J ) ) )
98rexbidv 2864 . . 3  |-  ( j  =  J  ->  ( E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  j )  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) ) )
106, 9elab3 3218 . 2  |-  ( J  e.  { j  |  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  j ) }  <->  E. x  e. 
TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  J ) )
112, 10bitri 249 1  |-  ( J  e.  2ndc  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2439   E.wrex 2799   _Vcvv 3076   class class class wbr 4399   ` cfv 5525   omcom 6585    ~<_ cdom 7417   topGenctg 14494   TopBasesctb 18633   2ndcc2ndc 19173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-nul 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-sn 3985  df-pr 3987  df-uni 4199  df-iota 5488  df-fv 5533  df-2ndc 19175
This theorem is referenced by:  2ndctop  19182  2ndci  19183  2ndcsb  19184  2ndcredom  19185  2ndc1stc  19186  2ndcrest  19189  2ndcctbss  19190  2ndcdisj  19191  2ndcomap  19193  2ndcsep  19194  dis2ndc  19195  tx2ndc  19355
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