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Theorem is2ndc 17188
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 17182 . . 3  |-  2ndc  =  { j  |  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x
)  =  j ) }
21eleq2i 2360 . 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 5555 . . . . 5  |-  ( topGen `  x )  e.  _V
53, 4syl6eqelr 2385 . . . 4  |-  ( ( x  ~<_  om  /\  ( topGen `
 x )  =  J )  ->  J  e.  _V )
65rexlimivw 2676 . . 3  |-  ( E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  J )  ->  J  e.  _V )
7 eqeq2 2305 . . . . 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 2577 . . 3  |-  ( j  =  J  ->  ( E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  j )  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) ) )
106, 9elab3 2934 . 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 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801   class class class wbr 4039   omcom 4672   ` cfv 5271    ~<_ cdom 6877   topGenctg 13358   TopBasesctb 16651   2ndcc2ndc 17180
This theorem is referenced by:  2ndctop  17189  2ndci  17190  2ndcsb  17191  2ndcredom  17192  2ndc1stc  17193  2ndcrest  17196  2ndcctbss  17197  2ndcdisj  17198  2ndcomap  17200  2ndcsep  17201  dis2ndc  17202  tx2ndc  17361
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  ax-nul 4165
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-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659  df-pr 3660  df-uni 3844  df-iota 5235  df-fv 5279  df-2ndc 17182
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