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Theorem is2ndc 17430
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 17424 . . 3  |-  2ndc  =  { j  |  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x
)  =  j ) }
21eleq2i 2451 . 2  |-  ( J  e.  2ndc  <->  J  e.  { j  |  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  j ) } )
3 simpr 448 . . . . 5  |-  ( ( x  ~<_  om  /\  ( topGen `
 x )  =  J )  ->  ( topGen `
 x )  =  J )
4 fvex 5682 . . . . 5  |-  ( topGen `  x )  e.  _V
53, 4syl6eqelr 2476 . . . 4  |-  ( ( x  ~<_  om  /\  ( topGen `
 x )  =  J )  ->  J  e.  _V )
65rexlimivw 2769 . . 3  |-  ( E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  J )  ->  J  e.  _V )
7 eqeq2 2396 . . . . 5  |-  ( j  =  J  ->  (
( topGen `  x )  =  j  <->  ( topGen `  x
)  =  J ) )
87anbi2d 685 . . . 4  |-  ( j  =  J  ->  (
( x  ~<_  om  /\  ( topGen `  x )  =  j )  <->  ( x  ~<_  om  /\  ( topGen `  x
)  =  J ) ) )
98rexbidv 2670 . . 3  |-  ( j  =  J  ->  ( E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  j )  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) ) )
106, 9elab3 3032 . 2  |-  ( J  e.  { j  |  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  j ) }  <->  E. x  e. 
TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  J ) )
112, 10bitri 241 1  |-  ( J  e.  2ndc  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2373   E.wrex 2650   _Vcvv 2899   class class class wbr 4153   omcom 4785   ` cfv 5394    ~<_ cdom 7043   topGenctg 13592   TopBasesctb 16885   2ndcc2ndc 17422
This theorem is referenced by:  2ndctop  17431  2ndci  17432  2ndcsb  17433  2ndcredom  17434  2ndc1stc  17435  2ndcrest  17438  2ndcctbss  17439  2ndcdisj  17440  2ndcomap  17442  2ndcsep  17443  dis2ndc  17444  tx2ndc  17604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-nul 4279
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-sn 3763  df-pr 3764  df-uni 3958  df-iota 5358  df-fv 5402  df-2ndc 17424
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