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Theorem is2ndc 17172
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 17166 . . 3  |-  2ndc  =  { j  |  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x
)  =  j ) }
21eleq2i 2347 . 2  |-  ( J  e.  2ndc  <->  J  e.  { j  |  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  j ) } )
3 simpr 447 . . . . 5  |-  ( ( x  ~<_  om  /\  ( topGen `
 x )  =  J )  ->  ( topGen `
 x )  =  J )
4 fvex 5539 . . . . 5  |-  ( topGen `  x )  e.  _V
53, 4syl6eqelr 2372 . . . 4  |-  ( ( x  ~<_  om  /\  ( topGen `
 x )  =  J )  ->  J  e.  _V )
65rexlimivw 2663 . . 3  |-  ( E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  J )  ->  J  e.  _V )
7 eqeq2 2292 . . . . 5  |-  ( j  =  J  ->  (
( topGen `  x )  =  j  <->  ( topGen `  x
)  =  J ) )
87anbi2d 684 . . . 4  |-  ( j  =  J  ->  (
( x  ~<_  om  /\  ( topGen `  x )  =  j )  <->  ( x  ~<_  om  /\  ( topGen `  x
)  =  J ) ) )
98rexbidv 2564 . . 3  |-  ( j  =  J  ->  ( E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  j )  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) ) )
106, 9elab3 2921 . 2  |-  ( J  e.  { j  |  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  j ) }  <->  E. x  e. 
TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  J ) )
112, 10bitri 240 1  |-  ( J  e.  2ndc  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   class class class wbr 4023   omcom 4656   ` cfv 5255    ~<_ cdom 6861   topGenctg 13342   TopBasesctb 16635   2ndcc2ndc 17164
This theorem is referenced by:  2ndctop  17173  2ndci  17174  2ndcsb  17175  2ndcredom  17176  2ndc1stc  17177  2ndcrest  17180  2ndcctbss  17181  2ndcdisj  17182  2ndcomap  17184  2ndcsep  17185  dis2ndc  17186  tx2ndc  17345
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  ax-nul 4149
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-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219  df-fv 5263  df-2ndc 17166
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