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Theorem is2ndc 20073
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 20067 . . 3  |-  2ndc  =  { j  |  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x
)  =  j ) }
21eleq2i 2535 . 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 5882 . . . . 5  |-  ( topGen `  x )  e.  _V
53, 4syl6eqelr 2554 . . . 4  |-  ( ( x  ~<_  om  /\  ( topGen `
 x )  =  J )  ->  J  e.  _V )
65rexlimivw 2946 . . 3  |-  ( E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  J )  ->  J  e.  _V )
7 eqeq2 2472 . . . . 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 2968 . . 3  |-  ( j  =  J  ->  ( E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  j )  <->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  J ) ) )
106, 9elab3 3253 . 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 1395    e. wcel 1819   {cab 2442   E.wrex 2808   _Vcvv 3109   class class class wbr 4456   ` cfv 5594   omcom 6699    ~<_ cdom 7533   topGenctg 14855   TopBasesctb 19525   2ndcc2ndc 20065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-sn 4033  df-pr 4035  df-uni 4252  df-iota 5557  df-fv 5602  df-2ndc 20067
This theorem is referenced by:  2ndctop  20074  2ndci  20075  2ndcsb  20076  2ndcredom  20077  2ndc1stc  20078  2ndcrest  20081  2ndcctbss  20082  2ndcdisj  20083  2ndcomap  20085  2ndcsep  20086  dis2ndc  20087  tx2ndc  20278
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